ON A HARDY SPACE APPROACH TO THE ANALYSIS OF SPECTRAL FACTORS

P. A. Fuhrmann^{∗}^{†}

Department of Mathematics Ben-Gurion University of the Negev Beer Sheva, Israel and A. Gombani^{‡} LADSEB-CNR Corso Stati Uniti 4 35127 Padova, ITALY

To appear Int. J. Contr.

February 18, 1998

Abstract

The paper presents a study of several problems related to spectral factorizations. We assume only a very weak form of coercivity for the p×p spectral function Φ and look at the set W^{m} of all rectangular, p × m spectral factors. The main object is the arithmetization of the geometry of the set of minimal, stable spectral factors by employing Hardy space techniques and the arithmetic of inner functions. Particular attention is paid to the study of various partial orders associated with the set of spectral factors.

SF/zpaper.tex

∗Earl Katz Family Chair in Algebraic System Theory †Partially supported by GIF under Grant No. I 184.

‡Supported by CNR

1 Introduction

The object of this paper is to present a study of the spectral factorization problem in the multivariable case. We will consider rational spectral function Φ of dimension p × p, of rank m_{0} ≤ p and degree 2n and we will characterize all the minimal, in the sense of McMillan degree, spectral factors of a given size p × m in the spectral domain, removing all constraints on rank and zero location. In particular only very weak coercivity assumptions will be made.

The topic of this paper has a long history, with roots in stochastic theory. We mention Anderson [1973], Faurre, Clerget and Germain [1979], Ruckebusch [1980], and in particular the series of seminal articles of Lindquist and Picci [1979abc,1985,1991] and Lindquist, Michaletzky and Picci [1994] as well as some papers by Pavon which have been the main inspiration of our work. So this problem has been widely studied in the existing literature. Still, a particular aspect of the problem seems to have escaped all these investigations and this was the motivation for starting this work. So, for example, it is well known how to characterize all the spectral factors in terms of a Riccati inequality (see Anderson [1973] and Faurre [1974]); nevertheless, a state space approach does not allow a precise characterization of the spectral factors of fixed dimension m where the usual constraints (m_{0} ≤ m ≤ p+n) are satisfied (these constraints come in a very natural manner from the Positive Real Lemma: we refer the reader to Anderson [1971] for the details). In Lindquist and Picci [1979c] an alternative spectral domain characterization is given for non full-rank Φ, but only for the internal case. In Lindquist Picci [1995] the external, but full-rank problem is treated. In conclusion, the sets of external factors of given size m of a non full-rank density Φ (which we will call W^{m}) are not very well characterized even today. Some properties of these sets are known (for example it is known it is a manifold for each m, see Baratchart and Gombani [1994]), but the underlying structure of these sets was quite unclear. The reasons for understanding such a problem are quite practical and come from time series analysis and multichannel signal transmission. The first problem occurs in econometrics, when a time series with a lot of variables should be explained in terms of the dynamics of few variables (factors) and some (white) disturbance error. The second occurs in decoding the signals which come to a cellular phone from the all the antennas which are operating on that cell and are of course broadcasting also to other phones. It can be shown that both situations can be brought to a nonlinear minimization problem on W^{m} (and the non full-rank assumption on Φ is crucial), and this explain why it is interesting to know that it is manifold and why, more generally, it is important to understand its structure. We do so in continuous time (even if the problems we just mentioned have a discrete time formulation), because of some simplified features of this setting. The discrete time version can be obtained by a suitable version of the Cayley transform.

With this problem in mind, we were led in a natural way to an approach which differs from that of Lindquist and Picci. Since we have removed the rank constraints we obtain some technical results which are more general than those previously obtained. The main difference however is one of emphasis. In the Lindquist-Picci approach to stochastic realization theory, geometry is reigning supreme. The approach is abstract with all the corresponding advantages and disadvantages. The first big advantage is that working with a space directly constructed from the stochastic process, stochastic objects like the past and future spaces are conceptually clearly defined and are independent of any representation. The disadvantages arise from the difficulties in the computational process: since every subspace has its own functional representation, the geometry of the stochastic setting is lost in the frequency domain. For better or worse, we do away with all that. The basic idea is to map the stochastic domain with a single isomorphism to the frequency domain. With each spectral factor we get corresponding representation spaces which are subspaces of a fixed vectorial L^{2} space of the imaginary axis. For normalization purposes we single out the minimal, stable, maximum phase spectral factor W_{+}. We study all other spectral factors in terms of their relation to W_{+}. The big advantage from our point of view is that the functional representations of these spaces in terms of inner functions, obtained using Beurling’s theorem, are all geometrically correlated (the Hilbert space structure is preserved). This allows us to replace the abstract geometrical constructs by the arithmetic properties of inner functions, making computations much easier. A second important feature of our approach that it solves the problem we considered in the beginning, namely the characterization of W^{m}.

The paper is structured as follows. We introduce some notations in section 2. In section 3, given a spectral function, we study the four minimal, extremal spectral factors. These factors are completely characterized by the requirement that all their zeros and poles lie in either the open left or right halfplanes. Section 4 is devoted to a study of Toeplitz operators whose symbol is a quotient of inner function. This class of Toeplitz operators is important because of the many connections to geometry. For some different applications to system theoretic problems, see Fuhrmann and Helmke [1997].

In section 5 we extend the scene by looking at the set of all minimal, stable rectangular spectral factors of a given size. Here the ground is laid for the use of factorization theory in the analysis of spectral factors. This is done via the use of the algebra of inner functions. The fine structure of the various factorizations associated with a given, stable spectral factor is analysed in Theorem 5.3. Since, via Beurling’s theorem, inner functions are closely related to the geometry of invariant subspaces in Hardy spaces, this leads to many geometric relations. These connections are studied in depth, as well as various associated projection operators. Some of this is summarized in Proposition 5.4. Special attention is given to two extreme cases, those of internal and external factors and characterizations of these factors are given. This study is closely related to geometric control theory, however the results obtained are beyond the scope of this paper. These results are relegated to future publications, see Gombani and Fuhrmann [1997] and Fuhrmann [1997].

We pass on, in section 6 to the study of a functional representation of the set of all minimal Markovian splitting subspaces, a key element in stochastic realization theory. With each minimal, stable spectral factor we associate a canonical state space, which serves also as the state space of a shift based realization. This allows us to link the geometry of splitting subspaces with the arithmetic properties of inner functions. One of the principal themes is to study a partial order relation in the set of all minimal, stable spectral factors. We begin, in Theorem 6.1, by studying the standard case of square, nonsingular spectral factors. While it does not generalize trivially to the singular, rectangular case, it indicates the direction. In keeping with the spirit of this paper, which is functional oriented, we avoid the use of the Riccati equation and inequality. This theme will be picked up in a different publication. Once the various partial orders, related to spectral factors, are introduced, we proceed to show the equivalence of these orders. This is the heart of this section and one of the principal results of the whole paper.

This is summarized in Theorem 6.2. We proceed, in Theorem 6.3, to characterize, functionally, the various spaces that appear in the scattering approach to stochastic realization theory. We conclude by studying some lattice properties of the set of all p × m stable, spectral factors. We note however that this set is not a complete lattice. The set of all minimal Markovian splitting subspaces spans a canonical subspace of H_{+}^{2}. We give a characterization of this space in section 7.

The reader who studies this paper will notice immediately that there are many important topics that have been omitted. In particular, little emphasis has been placed on state space techniques. Also, topics relating to geometric control theory, in the context of Hardy spaces, had to be avoided. The reason for this is simple. Inclusion of these topics, important as they are, would have been beyond the scope of a single paper. As indicated above, these topics will be treated in subsequent papers.

In a sense, the content of this paper is an attempt to understand stochastic realization theory in the style of Lindquist and Picci from a different point of view. Our debt to their work is evident throughout.

2 Preliminaries and notation

We work in the Hilbert space setting of the plane; we define, see Hoffman [1962], L^{2}( II ) to be the set of the vector or matrix valued, the proper dimension will be clear from the context, square integrable functions on the imaginary axis, and H_{+}^{2} to be the subspace of L^{2} of functions that are the boundary values of analytic functions in the right half-plane and such that

sup_{x>0} \frac{1}{2π} ∫_{−∞}^{∞} tr(F^{∗}(x + iy)F (x + iy)) dy < ∞

where ^{∗} denotes transposed conjugate. If F and G are column vectors, the inner product in H_{+}^{2} is

⟨F, G⟩ = \frac{1}{2π} ∫_{−∞}^{∞} G^{∗}(iω) F (iω) dω

Analogously, H_{+}^{∞} is the subspace of L^{2} of functions analytic in the right half-plane and such that

sup_{Re s>0} ∥F (s)∥ < ∞

where ∥F (x + iy)∥ denotes the usual matrix norm. (H_{−}^{2} and H_{−}^{∞} are defined similarly on the left half-plane).

Let F be function of L^{2}: we denote by P_{+}F (P_{−}F) the orthogonal projection of L^{2} onto H_{+}^{2}(H_{−}^{2}).

An p × m matrix valued function R̂ ∈ H_{+}^{2} is said to be row rigid if R̂R̂^{∗} = I_{p} (clearly this entails p ≤ m. It is column rigid if p ≥ m and R̂^{∗}R̂ = I_{m}. A function Q ∈ H_{+}^{∞} is inner if it is square and Q^{∗}Q = QQ^{∗} = I. It is well known that a column vector space M in H_{+}^{2} is invariant under multiplication by e^{−iωt} for t ≥ 0 if and only if it is of the form M = QH_{+}^{2} (Beurling’s Theorem). Similarly, a row space N is invariant if and only if it is of the form N = H_{+}^{2}Q. We set H_{c}(Q) := \{QH_{+}^{2}\}^{⊥} and H_{r}(Q) := \{H_{+}^{2}Q\}^{⊥}. Analogously, we define \overline{H}_{c}(Q^{∗}) := \{Q^{∗}H_{−}^{2}\}^{⊥} and \overline{H}_{r}(Q^{∗}) := \{H_{−}^{2}Q^{∗}\}^{⊥}, the orthogonal complement in these cases taken in H_{−}^{2}. Clearly, we have P_{+}e^{iωt}H_{c}(Q) ⊂ H_{c}(Q) and similarly for H_{r}(Q). A full column-rank p × m rational matrix function G in H_{+}^{∞} is said to be minimum-phase or outer (on the right) if rank G(s) = m for Re s > 0. It is well known that a rational function F in H_{+}^{∞} admits an outer inner factorization

F = F_{0}Q.

This factorization is unique up to a unitary constant matrix. For any W in L^{2} or in L^{∞} we define W^{∗}(s) := \overline{W (s̄)}^{T}, where¯ denotes conjugation and ^{T} denotes transposition. It should be noted that for an inner function Q^{∗} = Q^{−1}.

We say that an inner function Q_{1} divides Q_{2} on the right (left) if Q_{1}^{∗}Q_{2} ∈ H_{+}^{∞} (Q_{2}Q_{1}^{∗} ∈ H_{+}^{∞}). Given two inner functions Q_{1} and Q_{2} we denote the greatest common right (left) divisor by Q_{1} ∧_{R} Q_{2} (Q_{1} ∧_{L} Q_{2}) and the least common right (left) multiple by Q_{1} ∨_{R} Q_{2} (Q_{1} ∨_{L} Q_{2}). Two inner matrices are right (left) coprime if their greatest common right (left) divisor is the identity.

Two m × m inner matrices X and Y are equivalent if there exist inner matrices U, V such that X ∧_{L} V = I and Y ∧_{R} U = I and XU = V Y. Given an inner matrix X, it can be shown that there exists an (essentially) unique diagonal inner matrix D = diag {d_{1}, ..., d_{m}} such that d_{i}|d_{i+i} for i = 1, ..., m − 1 which is similar to X. The inner factors d_{i} which are not identically 1 are called invariant factors of X.

Remark: To distinguish between operators on row and column spaces, we will denote the operations on columns in the usual manner, whereas operations on row spaces are denoted by writing the argument before the operator. For example, given an operator T acting in a row space X with a subspace V, we would write V |T for the restriction of T to V. Ker V |T would denote the kernel of this map. This notation is immediately applied in the following definition.

Definition 2.1 Let G be a p × m matrix function in L^{∞}. We define the Hankel operator H_{G}^{c} acting on the column Hardy space as

\begin{matrix}H_{G}^{c} : H_{+}^{2} ⟶ H_{−}^{2} \\ f ↦ P_{+}Gf. \end{matrix}  (1)

Similarly, we define the Hankel operator H_{G}^{r} acting on row Hardy space by

\begin{matrix}H_{G}^{r} : H_{+}^{2} ⟶ H_{−}^{2} \\ f ↦ fGP_{+}. \end{matrix}  (2)

The function G is called the symbol of the Hankel operator. Similarly, conjugate Hankel operators are defined, with the role of H_{+}^{2} and H_{−}^{2} interchanged. We will use the notation Ĥ_{G}^{r} and Ĥ_{G}^{c} for these operators.

From now on we assume all the functions to be rational. We denote a realization of a

rational function W as ( \begin{matrix}A & B \\ C & D \end{matrix} ). With A^{#} we denote the Moore-Penrose inverse of a matrix

A.

3 Extremal spectral factors

Our object is to study the set of all minimal, not necessarily square, spectral factors of a given spectral function.

We assume we are given a rational spectral function Φ, that is a p×p proper rational matrix function which is nonnegative on the imaginary axis. We do not assume, as is usual, that Φ is regular on the imaginary axis. We do assume however that it is weakly coercive, namely that Φ has constant rank, m_{0} ≤ p, on the extended imaginary axis, i.e. including at the point of infinity. Furthermore, we assume that Φ, which clearly satisfies Φ(s) = Φ(−\overline{s})^{∗}, has McMillan degree 2n.

A p × m proper rational matrix function W is called a spectral factor if Φ = W W^{∗}. Here, as elsewhere W^{∗}(s) = W (−\overline{s})^{∗}. A spectral factor W is called stable (antistable) if W ∈ H_{+}^{∞} (W ∈ H_{−}^{∞}).

It turns out that the study of the set of minimal spectral factors is facilitated if we study it in relation to four extremal spectral factors. These four spectral factors are determined by the requirement that all their poles are located in either the left or right half planes, and the same for the zeros.

It is well known that spectral factors exist. Moreover, using the Beurling-Lax-Halmos theorem, there exists a stable, minimum phase, or outer spectral factor, which we denote by W_{−}. Our weak coercivity assumption implies actually that W_{−} is left invertible over H_{+}^{∞}. In a completely analogous way, there exists an essentially unique antistable and maximum phase spectral factor \overline{W}_{+}, which has also dimension p × m_{0}. By the same argument as before \overline{W}_{+} has an antistable left inverse.

Let K_{+} and K_{−} be the minimal essentially unique, that is unique up to a right constant unitary factor, m_{0} × m_{0} inner functions for which W_{+} = \overline{W}_{+}K_{+} is stable and \overline{W}_{−} = W_{−}K_{−}^{∗} is antistable. \overline{W}_{−} is the minimal, antistable minimum phase spectral factor and W_{+} is the minimal, stable maximum phase spectral factor, and the above factorizations are just the Douglas, Shapiro, Shields (DSS) factorizations of W_{+} and \overline{W}_{−} over H_{−}^{∞} and H_{+}^{∞} respectively. For more on this see Fuhrmann [1981].

We will assume in the following that the multiplicity of K_{+} equals m_{0}. This basically means

that if W_{−} = ( \begin{matrix}A & B_{−} \\ C & D_{−} \end{matrix} ) then rank B_{−} = m_{0}. In other words, if we seek decompositions of the

form Φ = Φ̂ + Δ with Δ constant and Φ of rank lower than m_{0}, we rule out that the problem has trivial solutions, namely those given by internal spectral factors. For more on this, see Baratchart and Gombani [1994].

Proposition 3.1 Let W be a minimal, stable p × m spectral factor. Then there exists an essentially unique m_{0} × m row rigid function Q̂ for which

W = W_{−}Q̂.  (3)

Without loss of generality, we will assume Q̂(∞) = ( I 0 ).

Proof: We consider H_{+}^{2}W which is an invariant subspace. Since rank W = m_{0} then, by Beurling’s theorem, there exists a rigid function Q̂ for which

H_{+}^{2}W = H_{+}^{2}Q̂.

Since W_{−} is outer, it follows that

H_{+}^{2}W = H_{+}^{2}Q̂ = H_{+}^{2}W_{−}Q̂.

This implies W = W_{−}Q̂.

The following proposition gives a characterization of the the image of all Hankel operators induced by minimal, stable, not necessarily square, spectral factors.

Proposition 3.2 Let W_{−} be the minimal, stable, minimum phase, spectral factor of a rational spectral function. Let W be any minimal, stable, not necessarily square, spectral factor. Let Ĥ_{W}^{c} : H_{−}^{2} ⟶ H_{+}^{2} be the associated conjugate Hankel operator. Then

1.

Im Ĥ_{W}^{c} = Im Ĥ_{W_{−}^{c}}.  (4)

In particular we have

Im Ĥ_{W_{+}}^{c} = Im Ĥ_{W_{−}}^{c}.  (5)

2. Let

W_{−} = KH_{−}^{∗} = (K^{∗})^{−1}H_{−}^{∗}  (6)

be a right coprime DSS factorization of W_{−} over H_{−}^{∞}. Let W be any stable,minimal spectral factor. Then it has a right coprime DSS factorization of the form

W = KH^{∗}.  (7)

Proof:

1. Let W be any minimal, stable, not necessarily square, spectral factor. We know, by Proposition 3.1, that there exists a row-rigid function Q̂ such that W = W_{−}Q̂ or, equivalently, that W_{−} = W Q̂^{∗}. By Lemma 3.5.7 in Fuhrmann [1981], we conclude that

Im Ĥ_{W_{−}}^{c} ⊂ Im Ĥ_{W}^{c} .  (8)

Now Im Ĥ_{W}^{c} can be taken, using the shift realization, as the state space of a minimal realization. Since W is, by assumption, a minimal spectral factor we have dim Im Ĥ_{W_{−}}^{c} = dim Im Ĥ_{W}^{c}. Thus (4) follows. Equality (5) is a special case.

2. We have Im Ĥ_{W}^{c} = {KH_{+}^{2}}^{⊥}. Since Im Ĥ_{W}^{c} = Im Ĥ_{W_{−}}^{c}, this implies a factorization, necessarily left coprime, of the form (7).

Equality (4) has practical implications for the realization of minimal spectral factors.

Corollary 3.1 Let

W_{−} = ( \begin{matrix}A & B_{−} \\ C & D_{−} \end{matrix} )  (9)

the realization being minimal and let W be any, not necessarily square, minimal stable spectral factor. Then W has a minimal realization of the form

W = ( \begin{matrix}A & B \\ C & D \end{matrix} )  (10)

Proof: We consider the shift realization, see Fuhrmann [1981,1996], based on the left coprime factorizations (6) and (7). Since the (C, A) operators in such a realization depend only on the inner function Q, the result follows. That (10) is a minimal realization follows from the minimality of W.

It might be tempting to use the previous result to try and study the set of all minimal, stable, spectral factors of given size by using the realizations with the same state space. This is not necessarily a good idea. In fact the main contribution of this paper comes from following a very different route, as will become apparent in section 6.

Proposition 3.3 Let W_{+} be the minimal, stable, maximum phase, spectral factor. If S is any nontrivial row rigid function, i.e. SS^{∗} = I, then

δ(W_{+}S) > δ(W_{+}).  (11)

Proof: Since W_{+}S is a stable spectral factor, and W_{+} = (W_{+}S)S^{∗}, we have

Im Ĥ_{W_{+}S}^{c} ⊃ Im Ĥ_{W_{+}}^{c}.  (12)

If δ(W_{+}S) = δ(W_{+}) then necessarily Im Ĥ_{W_{+}S}^{c} = Im Ĥ_{W_{+}}^{c}. By Proposition 3.1, there exist H^{∞} functions R, T such that

W_{+}S = W_{+}R^{∗} + T^{∗}.

Now W_{+} has an antistable left inverse W^{−L}, so S = R^{∗} +W^{−L}T^{∗}. Thus S is both in H_{+}^{∞} and in H_{−}^{∞} hence necessarily a constant coisometry, contrary to our assumption that S is nontrivial.

We proceed to derive a characterization of W_{+} which is analogous to Proposition 3.1. For this we will need the following proposition.

Proposition 3.4 Let

K_{1}^{−1}H_{1} = HK^{−1}  (13)

be coprime DSS factorizations over H_{+}^{∞}. Then if H_{1} is outer, so is H.

Proof: We prove it by contradiction. Assume H = E\overline{H} with E nontrivial. Let

( \begin{matrix}Y_{1} & −X_{1} \\ −H_{1} & K_{1} \end{matrix} ) ( \begin{matrix}K & X \\ H & Y \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

be a doubly coprime factorization. This implies also

( \begin{matrix}K & X \\ H & Y \end{matrix} ) ( \begin{matrix}Y_{1} & −X_{1} \\ −H_{1} & K_{1} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

In particular we get HY_{1} = Y H_{1} and hence E\overline{H}Y_{1} = Y H_{1}, which can be rewritten as \overline{H}Y_{1} = E^{−1}Y H_{1}. Now E, Y are left coprime as H, Y are. Let Ŷ Ê^{−1} be a right coprime factorization of E^{−1}Y. So

\overline{H}Y_{1} = E^{−1}Y H_{1} = Ŷ Ê^{−1}H_{1}.

Now the right coprimeness of Ŷ , Ê implies the existence of H^{∞} functions P, Q which solve the following Bezout identity

P Ŷ + QÊ = I.  (14)

This implies P Ŷ Ê^{−1}H_{1} +QH_{1} = Ê^{−1}H_{1}. Now L = P Ŷ Ê^{−1}H_{1} +QH_{1} = P \overline{H}Y_{1} +QH_{1} is clearly in H^{∞}, So H_{1} = ÊL contradicting the outerness of H_{1}.

Proposition 3.5 Let W_{+} be the minimal, stable, maximum phase, spectral factor. Let W_{+} = KH_{+}^{∗} be a left coprime DSS factorization over H_{−}^{∞}. Then H_{+} is outer.

Proof: Let \overline{W}_{+} be the minimal, maximum phase antistable spectral factor. Thus, in analogy to W_{−}, \overline{W}_{+} is conjugate outer, i.e. left invertible in H_{−}^{∞}. Let W_{+} be the minimal, maximum phase stable spectral factor. We have W_{+} = \overline{W}_{+}K_{+} = \overline{W}_{+}K_{+}^{−∗}, which is a right coprime DSS factorization over H_{−}^{∞}. Let W_{+} = K^{−∗}H_{+}^{∗} be a left coprime factorization of W_{+} over H_{−}^{∞}.

Now, by construction, \overline{W}_{+} is outer in H_{−}^{∞}, i.e. it is left invertible over H_{−}^{∞}. Applying Proposition 3.4 in its H_{−}^{∞} version, we conclude that H_{+}^{∗} is left invertible in H_{−}^{∞}. Equivalently, H_{+} is right invertible in H_{+}^{∞}, or it is row outer.

We proceed with the characterization of W_{+}.

Proposition 3.6 Let W be any minimal, stable not necessarily square spectral factor. Then there exists an m × m_{0} column rigid function Q̂′′ for which W Q̂′′ = W_{+}.

Proof: Let W_{+} = KH_{+}^{∗} and W = KH^{∗} be right coprime DSS factorizations. Since both W and W_{+} are spectral factors, we obtain

H^{∗}H = H_{+}^{∗}H_{+}.

Applying Proposition 3.1 in its H_{−}^{∞} form, we infer the existence of a row rigid function (Q̂′′)^{∗} ∈ H_{−}^{∞}, that is satisfying (Q̂′′)^{∗}Q̂′′ = I, for which H^{∗} = H_{+}^{∗}(Q̂′′)^{∗}. This implies

W = KH^{∗} = KH_{+}^{∗}(Q̂′′)^{∗} = W_{+}(Q̂′′)^{∗},

and hence W_{+} = W Q̂′′.

Proposition 3.7 There exist unique, m_{0} × m_{0} square inner functions Q_{−} and Q_{+} for which

\{ \begin{matrix}W_{+} & = & W_{−}Q_{+} \\ \overline{W}_{+} & = & \overline{W}_{−}Q_{−}. \end{matrix}  (15)

Moreover, we have

K_{−}Q_{+} = Q_{−}K_{+},  (16)

Q_{+}, K_{+} are right coprime and Q_{−}, K_{−} are left coprime.

Proof: W_{−}, \overline{W}_{+} are both of dimension p × m_{0}. Since \overline{W}_{−} = W_{−}K_{−}^{∗} and W_{+} = \overline{W}_{+}K_{+} with K_{−}, K_{+} inner of dimension m_{0} × m_{0}, it follows that W_{+}, \overline{W}_{−} are also of dimension p × m_{0}. By Proposition 3.1, there exists an m_{0}×m_{0} rigid, hence inner, function Q_{+} such that W_{+} = W_{−}Q_{+}. We easily compute that

W_{+} = \overline{W}_{−}K_{−}Q_{+} = \overline{W}_{−}Q_{−}K_{+}.

The left invertibility of \overline{W}_{−} implies (16).

That Q_{+} and K_{+} are right coprime follows from the fact that K_{+} is the minimal inner function for which \overline{W}_{+}K_{+} is stable. The second assertion is proved similarly.

A nontrivial right common factor for Q_{+}, K_{+} would contradict the fact that \overline{W}_{+} = W_{+}K_{+}^{∗} is a right coprime DSS factorization over H_{+}^{∞}. Similarly, for the left coprimeness of Q_{−} and K_{−}.

It is convenient to arrange the four spectral factors in the following diagram.

[figure: ^{\overline{W}^{−}}; ^{\overline{W}^{+}}; W_{+}; W_{−}; Q_{−}; Q_{+}; K_{−}; K_{+}]

Diagram 1

To get some intuition into the role of the inner functions Q_{±}, K_{±}, we recall that one convenient way to describe zeros and poles of rational functions is in terms of polynomial coprime factorizations.

Consider the minimal stable, minimum phase p × m_{0} spectral factor W_{−}. If we consider a polynomial left coprime factorization, then W_{−} = \overline{D}_{−}^{−1}\overline{E}_{−}. Here \overline{D}_{−} is a p×p stable nonsingular polynomial matrix, whereas \overline{E}_{−} is a p × m_{0} polynomial matrix all whose invariant factors are stable polynomials. The p×m_{0} numerator polynomial matrix admits a factorization EE_{−} with E a p × m_{0} polynomially left invertible matrix and E_{−} stable. Solving the polynomial spectral factorization problem

E_{−}E_{−}^{∗} = F_{+}F_{+}^{∗},  (17)

with F_{+} antistable and E_{−}^{−1}F_{+} biproper, we conclude that Q_{+} = E_{−}^{−1}F_{+} is an m_{0} × m_{0} inner function and

W_{−}Q_{+} = \overline{D}_{−}^{−1}\overline{E}_{−}(E_{−}^{−1}F_{+}) = \overline{D}_{−}^{−1}EE_{−}(E_{−}^{−1}F_{+}) = \overline{D}_{−}^{−1}(EF_{+}).

Obviously all the invariant factors of EF_{+} are antistable. So W_{−}Q_{+} = W_{+}, where W_{+} is the minimal stable maximum phase spectral factor.

Thus, the inner function Q_{+} describes the flipping of zeros from the open left half plane to the open right half plane. Given a factorization Q_{+} = Q_{1}Q_{2} into square inner factors, the left factor Q_{1} describes the set of all zeros of W_{−} flipped to the right half plane by Q_{1}, i.e. the set of antistable zeros of of the spectral factor W_{−}Q_{1}. Similarly, the right factor Q_{2} of Q_{+} describes the set of stable zeros of W = W_{−}Q_{1} = W_{+}Q_{2}^{∗} that remain to be flipped. The sets of zeros of this type are called internal zeros and the corresponding spectral factors are called internal spectral factors. We will discuss their role in more detail in Section 5.

As usual, we define the phase function T_{0} corresponding to the spectral function Φ by

T_{0} = \overline{W} _{+}^{−L}W_{−}.  (18)

Here, since \overline{W}_{+} is a p × m_{0} function which has full column rank on the extended imaginary axis, \overline{W} _{+}^{−L} denotes any left inverse. The phase function is an m_{0} × m_{0} all-pass function. It plays an important role in understanding the set of all minimal spectral factors.

The next lemma relates the inner functions K_{±}, Q_{±} to the phase function.

Lemma 3.1 Let W_{−}, W_{+}, \overline{W}_{−}, \overline{W}_{+} be the extremal spectral factors corresponding to the spectral function Φ and let T_{0} be the associated phase function. Then we have

\overline{W}_{+}\overline{W} _{+}^{−L}W_{+} = W_{+}.  (19)
\overline{W}_{+}\overline{W} _{+}^{−L}W_{−} = W_{−}.  (20)
W_{+}W_{+}^{−L}W_{−} = W_{−}.  (21)
T_{0} = \overline{W} _{+}^{−L}W_{−} = K_{+}Q_{+}^{−1} = Q_{−}^{−1}K_{−}.  (22)

Proof: Since \overline{W} _{+}^{−L}\overline{W}_{+} = I, we have \overline{W}_{+}\overline{W} _{+}^{−L}\overline{W}_{+} = \overline{W}_{+}. Now

\overline{W}_{+}\overline{W} _{+}^{−L}W_{+} = \overline{W}_{+}\overline{W} _{+}^{−L}\overline{W}_{+}K_{+} = \overline{W}_{+}K_{+} = W_{+}.  (23)

This proves (19). (20) follows, using the fact that W_{−}Q_{+} = W_{+}. Substituting W_{+} = W_{−}Q_{+} in the equality W_{+}W_{+}^{−L}W_{+} = W_{+} and eliminating Q_{+} leads to (21). Finally, using (21), we compute

T_{0} = \overline{W} _{+}^{−L}W_{−} = \overline{W} _{+}^{−L}(W_{+}W_{+}^{−L}W_{−}) = (\overline{W} _{+}^{−L}W_{+})(W_{+}^{−L}W_{−}) = K_{+}Q_{+}^{−1}.

That K_{+}Q_{+}^{−1} = Q_{−}^{−1}K_{−} was proved in (16).

As a result of this lemma, it is clear that the phase function is inner if and only if Q_{±} = I, i.e. if and only if there are no internal zeros. The principal difference from the regular case is that the degrees of the determinants of Q_{+} and K_{+} are no longer necessarily equal. This difference manifests itself geometrically, as well as in the invertibility properties of Toeplitz operators. This will be studied in section 4. The phase function is a powerful tool for the analysis of spectral factors. It turns out that for a finer analysis we need localized versions of the phase function corresponding to ordered pairs of internal spectral factors. We postpone the introduction of these functions until, in section 5, we have constructed the right notation for it.

We will find it convenient to deal with square inner functions. To this end we define the extended, extremal spectral factors, W_{−}^{e}, W_{+}^{e}, \overline{W} _{−}^{e}, \overline{W} _{+}^{e}, as the p × p matrix functions which are obtained by augmenting with p − m_{0} zero columns. For example W_{−}^{e} = ( W_{−} 0 ). Similarly, we extend the inner functions Q_{±}, K_{±} in an obvious way to m × m inner functions by letting

Q_{±}^{e} = ( \begin{matrix}Q_{±} & 0 \\ 0 & I \end{matrix} ) ,  K_{±}^{e} = ( \begin{matrix}K_{±} & 0 \\ 0 & I \end{matrix} ) .  (24)

Clearly, we obtain the commutativity of the following diagram

[figure: ^{\overline{W} {e \atop −}}; ^{\overline{W} {e \atop +}}; W_{+}^{e}; W_{−}^{e}; Q_{−}^{e}; Q_{+}^{e}; K_{−}^{e}; K_{+}^{e}]

Diagram 2

The previous coprimeness conditions extend also in a natural way to the extended inner functions. Thus Q_{+}^{e}, K_{+}^{e} are right coprime and Q_{−}^{e}, K_{−}^{e} are left coprime.

4 On Toeplitz operators

In this section we will turn our attention to a study of the phase function and some Toeplitz operators related to it. The invertibility of Toeplitz operators has both geometric and arithmetic characterizations. By arithmetic characterizations we refer to Wiener-Hopf factorizations and the corresponding factorization indices, whereas the geometry refers to the invariant and coinvariant subspaces of Hardy spaces.

We recall the definition of Toeplitz operators. As we are working with both row and column spaces we must distinguish between two types of Toeplitz operators.

Definition 4.1 Let G be an m × m matrix function in L^{∞}. We define the Toeplitz operator T_{G}^{c} acting on column Hardy space by

\begin{matrix}T_{G}^{c} : H_{+}^{2} ⟶ H_{+}^{2} \\ f ↦ P_{+}Gf. \end{matrix}  (25)

Similarly, we define the Toeplitz operator T_{G}^{r} acting on row Hardy space by

\begin{matrix}T_{G}^{r} : H_{+}^{2} ⟶ H_{+}^{2} \\ f ↦ fGP_{+}. \end{matrix}  (26)

In both cases G is called the symbol of the corresponding Toeplitz operator.

The adjoints of Toeplitz operators are also Toeplitz operators. In fact we have

\begin{matrix}(T_{G}^{r})^{∗} = T_{G^{∗}}^{r} \\ (T_{G}^{c})^{∗} = T_{G^{∗}}^{c} \end{matrix}

We say that

G = G_{−}Δ_{r}G_{+}  (27)

is a right Wiener-Hopf factorization if G_{−}^{±1} ∈ H_{−}^{∞}, G_{+}^{±1} ∈ H_{+}^{∞} and

Δ_{r}(s) = diag ((\frac{s − 1}{s + 1})^{κ_{1}} , …, (\frac{s − 1}{s + 1})^{κ_{m}})  (28)

with κ_{1} ≥ ⋯ ≥ κ_{m}. The indices κ_{1}, ⋯, κ_{m} are called the right Wiener-Hopf factorization indices. Left factorizations and left factorization indices are similarly defined.

It is well known that if G ∈ L^{∞} is continuous, and in particular if it is rational, then Wiener-Hopf factorization exist and the factorization indices, though not necessarily the factorizations, are uniquely defined.

Proposition 4.1 Let G ∈ L^{∞} and G = G_{−}Δ_{r}G_{+} its right Wiener-Hopf factorization. Then

1. The following statements are equivalent.

(a) The Toeplitz operator T_{G}^{c} is injective.

(b) The Toeplitz operator T_{Δ_{r}}^{c} is injective.

(c) Δ_{r} is full column rank and all the right Wiener-Hopf factorization indices are nonnegative.

2. The following statements are equivalent. (a) The Toeplitz operator T_{G}^{c} is surjective.

(b) The Toeplitz operator T_{Δ_{r}}^{c} is surjective.

(c) Δ_{r} is full row rank and all the right Wiener-Hopf factorization indices are nonpositive.

Proof: The maps T_{G_{−}}^{c}, T_{G_{−}^{−1}}^{c}, T_{G_{+}}^{c}, T_{G_{+}^{−1}}^{c} are all invertible. In fact (T_{G_{−}}^{c})^{−1} = T_{G_{−}^{−1}}^{c} and (T_{G_{+}}^{c})^{−1} = T_{G_{+}^{−1}}^{c}. Clearly we have the following commutative diagram

[figure: H_{+}^{2}; H_{+}^{2}; H_{+}^{2}; H_{+}^{2}; T_{Δ_{r}}^{c}; T_{G}^{c}; T_{G_{+}}^{c}; T_{G_{−}}^{c}]

Diagram

as

T_{G_{−}}^{c}T_{Δ_{r}}^{c}T_{G_{+}}^{c}f = P_{+}G_{−}P_{+}Δ_{r}G_{+}f = P_{+}G_{−}Δ_{r}G_{+}f = T_{G}^{c}f.

In particular this shows that T_{G}^{c} is injective if and only if T_{Δ_{r}}^{c} is injective. Now, if Δ_{r} is given by (28), then clearly

T_{Δ_{r}}^{c} = T(_{\frac{s−1}{s+1}})^{κ_{1}} ⊕ ⋯ ⊕ T(_{\frac{s−1}{s+1}})^{κm}.

So it suffices to analyse a Toeplitz operator of the form T(_{\frac{s−1}{s+1}})^{κ} operating on the scalar H_{+}^{2} space. Since \frac{s − 1}{s + 1} is inner, we have for κ ≥ 0 that T(_{\frac{s−1}{s+1}})^{κ} is isometric and hence injective. The codimension of the image is equal to κ. For κ ≤ 0 the operator T(_{\frac{s−1}{s+1}})^{κ} is surjective and

the dimension of the kernel is κ. In fact we have Ker T(_{\frac{s−1}{s+1}})^{κ} = \{(\frac{s−1}{s+1})^{−κ} H_{+}^{2}\}^{⊥}. This implies dim Ker T(_{\frac{s−1}{s+1}})^{κ} = −κ.

Going back to the operator T_{Δ_{r}}^{c} we obtain

dim Ker T_{Δ_{r}}^{c} = − ∑_{κ_{i}<0} κ_{i}
codim Im T_{Δ_{r}}^{c} = ∑_{κ_{i}>0} κ_{i}.

Thus injectivity is equivalent to all the right Wiener-Hopf indices being nonnegative and surjectivity to all the right Wiener-Hopf indices being nonpositive.

In view of the preceeding proposition, the following corollary is obvious.

Corollary 4.1 1. The Toeplitz operator T_{G}^{c} is invertible if and only if G is invertible and all the right Wiener-Hopf factorization indices are trivial.

2. The Toeplitz operator T_{G}^{r} is invertible if and only if G is invertible and all the left Wiener-Hopf factorization indices are trivial.

The following proposition connects invertibility properties of Toeplitz operators acting in row and column Hardy spaces.

Proposition 4.2 Let G ∈ L^{∞} be rational. Then

1. The Toeplitz operator T_{G}^{c} is injective if and only if T_{G^{∗}}^{r} is surjective.

2. The Toeplitz operator T_{G}^{c} is surjective if and only if T_{G^{∗}}^{r} is injective.

3. The Toeplitz operator T_{G}^{c} is injective if and only if T_{G}^{r} is injective.

Proof:

1. Let G = G_{−}Δ_{r}G_{+} be a right Wiener-Hopf factorization. Then G^{∗} = G_{+}^{∗}Δ_{r}^{∗}G_{−}^{∗} is a left Wiener-Hopf factorization of G^{∗}. The factorization indices of G are all nonnegative if and only if the factorization indices of G^{∗} are all nonpositive.

2. The proof is similar.

3. Since (T_{G}^{c})^{∗} = T_{G^{∗}}^{c}, T_{G}^{c} is injective if and only if T_{G^{∗}}^{r} is surjective. By part (2) this is the case if and only if T_{G}^{r} is injective.

Note that, if G = G_{−}Δ_{r}G_{+} is a right Wiener-Hopf factorization with nonnegative factorization indices, then we have an explicit formula for the left inverse of T_{G}^{c}, namely

(T_{G}^{c})^{−L}g = G_{+}^{−1}P_{+}Δ^{−1}G_{−}^{−1}g.

We do not have such a formula for the right inverse of T_{G}^{r} based on the same right Wiener-Hopf factorization.

The principal tool in our analysis are Theorem 4.1 and Proposition 4.4. Most of the content can be found in Nikolskii [1985]. The connection with Wiener-Hopf factorization indices is a direct application of well known results in the study of Toeplitz operators, see for example Gohberg and Feldman [1971]. In the proofs we need the following simple lemma.

Lemma 4.1 1. Let V, W be two subspaces of a Hilbert space H, with dim W < ∞. Let P_{V} be the orthogonal projection of H on V. Then

P_{V} W = V

if and only if

H = V^{⊥} + W.

2. Let U, V, W be subspaces of a Hilbert space H. If W is orthogonal to both U and V, then

U ∩ V = U ∩ (V + W).  (29)

Proof:

1. If H = V^{⊥} + W, then clearly V = P_{V} H = P_{V} (V^{⊥} + W) = P_{V} W.

Conversely, assume P_{V} W = V. Let x ∈ (V^{⊥} + W)^{⊥} = V ∩ W^{⊥}. Let w ∈ W be such that P_{V} w = x, then necessarily (x, w) = 0. Therefore

0 = (x, w) = (P_{V} w, w) = ∥P_{V} w∥^{2} = ∥x∥^{2}.

So x = 0, i.e. \overline{V^{⊥} + W} = H. Since W is finite dimensional, we have also V^{⊥} + W = H.

2. Clearly U ∩V ⊂ U ∩(V +W). Assume conversely that f ∈ U ∩(V +W), i.e. f = u = v+w. Since w = u − v, we have w ∈ (U + V) ∩ (U^{⊥} ∩ V^{⊥}) = (U + V) ∩ (U + V)^{⊥}. So necessarily w = 0 and u = v, i.e. f ∈ U ∩ V.

Proposition 4.3 Let Q, \overline{Q}, K_{1}, K_{2} be rational inner functions satisfying

K_{1}Q = \overline{Q}K_{2}.  (30)

with K_{1}, \overline{Q} left coprime and Q, K_{2} right coprime. Then

dim Ker T_{QK_{2}^{∗}}^{r} = dim H_{r}(K_{1})Q ∩ H_{r}(K_{2}) = dim H_{+}^{2}Q ∩ H_{r}(K_{2}).  (31)

Specifically, the map g ↦ gQ is a bijective linear map from Ker T_{QK_{2}^{∗}}^{r} to H_{r}(K_{1})Q ∩ H_{r}(K_{2}).

We have

dim Ker T_{QK_{2}^{∗}}^{r} = dim{f | ∥H_{QK_{2}^{∗}}^{r}f∥ = ∥f∥}.

Proof: Assume f ∈ H_{r}(K_{1})Q ∩ H_{r}(K_{2}), i.e. f = g_{1}Q = g_{2} = h_{2}K_{2} with g_{1} ∈ H_{r}(K_{1}), g_{2} ∈ H_{r}(K_{2}) and h_{2} ∈ \overline{H}_{r}(K_{2}^{∗}). Thus g_{1}QK_{2}^{∗} = h_{2}. Applying the orthogonal projection P_{+}, we obtain g_{1}QK_{2}^{∗}P_{+} = h_{2}P_{+} = 0, i.e. g_{1} ∈ Ker T_{QK_{2}^{∗}}^{r}.

Conversely, assume g_{1} ∈ Ker T_{QK_{2}^{∗}}^{r}, i.e. g_{1}QK_{2}^{∗}P_{+} = 0. This means that g_{1}QK_{2}^{∗} = h_{2} with h_{2} ∈ H_{−}^{2}. From the equality f = g_{1}Q = h_{2}K_{2} we conclude that f ∈ H_{+}^{2}Q ∩ H_{r}(K_{2}). Now, using the equality K_{1}Q = \overline{Q}K_{2}, we can write H_{+}^{2} = H_{+}^{2}K_{1} ⊕ H_{r}(K_{1}) and hence

H_{+}^{2}Q = H_{+}^{2}K_{1}Q ⊕ H_{r}(K_{1})Q = H_{+}^{2}\overline{Q}K_{2} ⊕ H_{r}(K_{1})Q.

Obviously H_{+}^{2}\overline{Q}K_{2} ⊥ H_{r}(K_{2}), and hence, applying Lemma 4.1 (ii), we get H_{r}(K_{1})Q∩H_{r}(K_{2}) = H_{+}^{2}Q ∩ H_{r}(K_{2}).

For the notation used in the following theorem, as well as in the rest of the paper, the reader is advised to consult the remark preceeding Definition 2.1.

Theorem 4.1 Let Q_{−}, Q_{+}, K_{−}, K_{+} be rational inner functions satisfying

Q_{−}K_{+} = K_{−}Q_{+}.  (32)

and the coprimeness conditions

\begin{aligned}K_{−} ∧_{L} Q_{−} & = & I \\ K_{+} ∧_{R} Q_{+} & = & I. \end{aligned}  (33)

Then

1. The following statements are equivalent

(a) The Toeplitz operator T_{K_{+}Q_{+}^{∗}}^{r} = T_{Q_{−}^{∗}K_{−}}^{r} is injective.

(b) The Toeplitz operator T_{Q_{+}K_{+}^{∗}}^{r} = T_{K_{−}^{∗}Q_{−}}^{r} is surjective.

(c) All left Wiener-Hopf factorization indices of K_{+}Q_{+}^{∗} = Q_{−}^{∗}K_{−} are nonnegative.

(d) We have

Ker H_{r}(Q_{+})|P_{H_{r}(K_{+})} = H_{r}(Q_{+}) ∩ H_{+}^{2}K_{+} = {0}  (34)

(e) We have

H_{+}^{2}Q_{+} + H_{−}^{2}K_{+} = L^{2}.  (35)

(f) We have

H_{+}^{2}K_{+} ∩ H_{−}^{2}Q_{+} = {0}.  (36)

(g) We have

H_{+}^{2}Q_{+} + H_{r}(K_{+}) = H_{+}^{2}.  (37)

(h) We have

H_{r}(K_{+})P_{H_{r}(Q_{+})} = H_{r}(Q_{+}).  (38)

(i) All the singular values of the Hankel operator H_{K_{+}Q_{+}^{∗}}^{r} = H_{K_{−}^{∗}Q_{−}}^{r} are < 1.

2. The following statements are equivalent

(a) The Toeplitz operator T_{K_{+}Q_{+}^{∗}}^{c} = T_{Q_{−}^{∗}K_{−}}^{c} is injective.

(b) The Toeplitz operator T_{Q_{+}K_{+}^{∗}}^{c} = T_{K_{−}^{∗}Q_{−}}^{c} is surjective.

(c) All right Wiener-Hopf factorization indices of Q_{+}K_{+}^{∗} = K_{−}^{∗}Q_{−} are nonnegative.

(d) We have

Ker P_{H_{c}(K_{−})}|H_{c}(Q_{−}) = H_{c}(Q_{−}) ∩ K_{−}H_{+}^{2} = {0}  (39)

(e) We have

H_{c}(K_{−}) + Q_{−}H_{+}^{2} = H_{+}^{2}.  (40)

(f) We have

K_{−}H_{−}^{2} + Q_{−}H_{+}^{2} = L^{2}.  (41)

(g) We have

P_{H_{c}(Q_{−})}H_{c}(K_{−}) = H_{c}(Q_{−}).  (42)

(h) All the singular values of the Hankel operator H_{K_{+}Q_{+}^{∗}}^{c} = H_{K_{−}^{∗}Q_{−}}^{c} are < 1.

3. The following statements are equivalent (a) The Toeplitz operator T_{K_{+}Q_{+}^{∗}}^{c} = T_{K_{−}^{∗}Q_{−}}^{c} is surjective.

(b) The Toeplitz operator T_{Q_{+}K_{+}^{∗}}^{c} = T_{K_{−}^{∗}Q_{−}}^{c} is injective.

(c) All right Wiener-Hopf factorization indices of Q_{+}K_{+}^{∗} = K_{−}^{∗}Q_{−} are nonnegative.

(d) We have

Ker P_{H_{c}(Q_{−})}|H_{c}(K_{−}) = H_{c}(K_{−}) ∩ Q_{−}H_{+}^{2} = {0}  (43)

(e) We have

K_{−}H_{+}^{2} + Q_{−}H_{−}^{2} = L^{2}.  (44)

(f) We have

K_{−}H_{+}^{2} + H_{c}(Q_{−}) = H_{+}^{2}.  (45)

(g) We have

P_{H_{c}(K_{−})}H_{c}(Q_{−}) = H_{c}(K_{−}).  (46)

(h) All the singular values of the Hankel operator H_{K_{+}Q_{+}^{∗}}^{c} = H_{K_{−}^{∗}Q_{−}}^{c} are < 1.

4. The following statements are equivalent (a) The Toeplitz operators T_{Q_{+}K_{+}^{∗}}^{c} = T_{K_{−}^{∗}Q_{−}}^{c} and T_{Q_{+}K_{+}^{∗}}^{r} = T_{K_{−}^{∗}Q_{−}}^{r} are both invertible.

(b) All, left and right, Wiener-Hopf factorization indices of Q_{+}K_{+}^{∗} = K_{−}^{∗}Q_{−} are trivial.

(c) We have

\{ \begin{matrix}H_{c}(K_{−}) ∩ Q_{−}H_{+}^{2} & = & \{0\} \\ H_{c}(K_{−}) + Q_{−}H_{+}^{2} & = & H_{+}^{2} \end{matrix}  (47)

(d) We have

\{ \begin{matrix}P_{H_{r}(K_{−})}H_{r}(Q_{−}) & = & H_{r}(K_{−}) \\ P_{H_{c}(Q_{−})}H_{c}(K_{−}) & = & H_{c}(Q_{−}) \end{matrix}  (48)

(e) We have

\{ \begin{matrix}K_{−}H_{−}^{2} ∩ Q_{−}H_{+}^{2} & = & \{0\} \\ K_{−}H_{−}^{2} + Q_{−}H_{+}^{2} & = & L^{2} \end{matrix}  (49)

(f) All singular values of the Hankel operators H_{Q_{+}K_{+}^{∗}}^{c} are less than 1.

dim Ker T_{Q_{+}K_{+}^{∗}}^{c} = dim{f | ∥H_{Q_{+}K_{+}^{∗}}^{c}f∥ = ∥f∥}.  (50)

Proof:

1. The equivalence of (a)-(c) follows from Proposition 4.1.

(a) ⇔ (d) We have f ∈ Ker T_{K_{+}Q_{+}^{∗}}^{r} if and only if 0 = fK_{+}Q_{+}^{∗}P_{+}, i.e. if and only if fK_{+} ∈ H_{−}^{2}Q_{+}, that is if and only if g = fK_{+} ∈ H_{−}^{2}Q_{+} ∩ H_{+}^{2}K_{+}. Since K_{+} is inner, the equivalence follows.

(e) ⇒ (g) Applying P_{+} to equality (35) yields (37).

(g) ⇒ (h) We apply the projection P_{H_{r}(Q_{+})} to equality (37) to get (38).

(h) ⇒ (a) Assume H_{r}(K_{+})|P_{H_{r}(Q_{+})} is surjective, i.e. H_{r}(Q_{+})|P_{H_{r}(K_{+})} is injective. This implies (34), and hence also (a).

(a) ⇔ (i)

Since for any L^{∞} matrix A and f ∈ H_{+}^{2} we have

∥fA∥^{2} = ∥fAP_{−}∥^{2} + ∥fAP_{+}∥^{2} = ∥fH_{A}^{r}∥^{2} + ∥fT_{A}^{r}∥^{2},

it is clear that if A is an all-pass function, i.e. isometric valued, then we have

∥f∥^{2} = ∥fH_{A}^{r}∥^{2} + ∥fT_{A}^{r}∥^{2}.

In particular we have f ∈ Ker T_{A}^{r} if and only if ∥f∥ = ∥fH_{A}^{c}∥. Specializing to the all-pass function K_{+}Q_{+}^{∗} = Q_{−}^{∗}K_{−}, we get

Ker T_{K_{+}Q_{+}^{∗}}^{r} = {f| ∥fH_{K_{+}Q_{+}^{∗}}^{r}∥ = ∥f∥}.

In particular T_{K_{+}Q_{+}^{∗}}^{r} is injective if and only if all the singular values of the Hankel operator H_{K_{+}Q_{+}^{∗}}^{r} = H_{Q_{−}^{∗}K_{−}}^{r} are < 1.

All other statement are proved analogously.

We would like to point out that, in view of 4.2, the statements in parts 1. and 3. of the previous theorem are equivalent. The equivalence of some of the relations can be proved directly. As an example we prove the equivalence of (41) and (35), using the equality (32) and the fact that multiplication by all pass functions is a unitary map in L^{2}.

\begin{aligned}L^{2} & = & K_{−}H_{−}^{2} + Q_{−}H_{+}^{2} = H_{−}^{2} + K_{−}^{∗}Q_{−}H_{+}^{2} \\ & = & H_{−}^{2} + Q_{+}K_{+}^{∗}H_{+}^{2} = Q_{+}^{∗}H_{−}^{2} + K_{+}^{∗}H_{+}^{2}. \end{aligned}

The equality L^{2} = Q_{+}^{∗}H_{−}^{2} + K_{+}^{∗}H_{+}^{2} for column spaces is clearly equivalent to the equality L^{2} = H_{+}^{2}Q_{+} + H_{−}^{2}K_{+} for row spaces.

One other thing worth noting is the close connection between Toeplitz operators and projection operators. In the context of polynomial models it was put to effective use in Fuhrmann [1981].

The previous theorem was essentially about a special class of Toeplitz operators, but did not relate to a spectral function or a spectral factorization problem. In that case, more can be said. Heuristically, based on our assumptions, the number of zeros of a spectral factor, as measured by Q_{±} cannot exceed the number of poles as measured by K_{±}. We state the result as a one sided invertibility of the Toeplitz operators with the symbol equal to the phase function.

Proposition 4.4 Let Φ be a p×p, rank m_{0}, weakly coercive, spectral function, and let W_{±}, \overline{W}_{±} be the p×m_{0} extremal spectral factors. Then both Toeplitz operators T_{\overline{W} _{+}^{−L}W_{−}}^{c} = T_{K_{+}Q_{+}^{∗}}^{c} = T_{Q_{−}^{∗}K_{−}}^{c} and T_{\overline{W} _{+}^{−L}W_{−}}^{r} are injective.

Proof: For the injectivity of T_{\overline{W} _{+}^{−L}W_{−}}^{c} we show an explicit left inverse, namely the map f ↦

W_{−}^{−L}P_{+}\overline{W}_{+}f. Indeed, using (20), we compute

\begin{aligned}W_{−}^{−L}P_{+}\overline{W}_{+}P_{+}\overline{W} _{+}^{−L}W_{−}f & = & W_{−}^{−L}P_{+}\overline{W}_{+}\overline{W} _{+}^{−L}W_{−}f \\ & = & W_{−}^{−L}P_{+}W_{−}f = W_{−}^{−L}W_{−}f = f. \end{aligned}

The injectivity of T_{\overline{W} _{+}^{−L}W_{−}}^{r} follows from Proposition 4.2.

Corollary 4.2 With the notation of Proposition 4.4, we have the following.

1. The maps H_{r}(Q_{+})|P_{H_{r}(K_{+})} and P_{H_{c}(K_{−})}|H_{c}(Q_{−}) are injective.

2. The maps H_{r}(K_{+})|P_{H_{r}(Q_{+})} and P_{H_{c}(Q_{−})}|H_{c}(K_{−}) are surjective.

Proof: Follows from Proposition 4.4 and Theorem 4.1. The second statement follows by duality.

5 Rectangular spectral factors

In this section we study the main object of the paper, namely the set W^{m} of all minimal stable, rectangular spectral factor of size p × m for m_{0} ≤ m ≤ p + n. In particular, we are interested in the parametrization of this set for a given m. We will study first two of these sets of minimal, stable rectangular spectral factors: those that are internal (i.e. such that m = m_{0}) and those that are external (in a sense that will be made precise later). The understanding of these, in a sense opposite, special cases enables us to fully understand the general rectangular case. Our approach is relating the set of spectral factors to the arithmetic of inner functions and their factorizations. This will also be interpreted later on geometrically.

As in the case of regular spectral factors, the four extremal spectral factors studied in section 3 provide the framework for our analysis of all rectangular spectral factors.

The next proposition studies the embedding of rigid functions in inner ones. This is a special case of Darlington synthesis (see Dewilde [1976]).

Proposition 5.1 1. Let R̂ be a p × m_{0} column rigid function, that is R̂^{∗}R̂ = I. Then there exists a p × (p − m_{0}) column rigid function R̃ such that

(a) R = ( R̂ R̃ ) is inner.

(b) We have the equality of McMillan degrees

δ(R) = δ(R̂).  (51)

R̃ is uniquely determined up to a right constant unitary factor.

2. Let R̂ be a m_{0} × p row rigid function, that is R̂R̂^{∗} = I. Then there exists a (p − m_{0}) × p row rigid function R̃ such that

(a) R = ( R̂ ; R̃ ) is inner.

(b) We have the equality of McMillan degrees

δ(R) = δ(R̂).  (52)

R̃ is uniquely determined up to a left constant unitary factor.

Proof:

1. Assume R̂^{∗}R̂ = I, then R̂R̂^{∗} ≤ I Assume an inner embedding of R̂ exists, i.e. there exists an R̃ such that R = ( R̂ R̃ ) is inner. Since R^{∗}R = RR^{∗} = I, we have R̃R̃^{∗} = I − R̂R̂^{∗} ≥ 0. Thus R̃ is a spectral factor of I − R̂R̂^{∗}. To minimize the McMillan degree of R, which is clearly bounded below by δ(R̂), we take R̃ to be the outer spectral factor of I − R̂R̂^{∗}.

We show now that in this case

Im HR̃ ⊂ Im HR̂.  (53)

This inclusion is equivalent to

ker H∗ ; R̂ ⊂ ker H∗ ; R̃.  (54)

Note that f ∈ ker H∗ ; R̂ if and only if R̂^{∗}f ∈ H_{+}^{2} and similarly f ∈ ker H∗ ; R̃ if and only if R̃^{∗}f ∈ H_{+}^{2}. If the inclusion (54) does not hold, there exists an f ∈ H_{+}^{2} such that R̂^{∗}f ∈ H_{+}^{2} and R̃^{∗}f = g + h, with g ∈ H_{+}^{2} and 0 ≠ h ∈ H_{−}^{2}. Now

f = ( R̂ R̃ ) ( R̂^{∗} ; R̃^{∗} ) f = R̂(R̂^{∗}f) + R̃(g + h),

which shows that there exists a nonzero h ∈ H_{−}^{2} for which R̃h ∈ H_{+}^{2} or, equivalently, R̃h ⊥ H_{−}^{2}. However this implies h ⊥ R̃^{∗}H_{−}^{2}. Since R̃ is row outer, we have H_{+}^{2}R̃ = H_{+}^{2} and hence R̃^{∗}H_{−}^{2} = H_{−}^{2}. This shows that h = 0, contradicting our assumption. Thus the inclusion (53) holds. This inclusion immediately implies

Im H_{( _{R̂ R̃} )} = Im HR̂,

and hence, by realization theory, that δ(( R̂ R̃ )) = δ(R̂).

2. The proof is similar or can be obtained from the first part by duality considerations.

We will refer to the extensions obtained in the previous proposition as minimal inner embeddings.

The next proposition relates the extremal factors to other minimal, stable and antistable, spectral factors via the use of inner functions. Since inner functions are, by Beurling’s theorem, intimately related to invariant subspaces, this result opens up the possibility of a geometric approach to the study of spectral factors.

Proposition 5.2 1. Let W_{−}^{e} and W_{+}^{e} be the extended, stable, minimum and maximum phase respectively, spectral factors. Given any minimal stable spectral factor W, there exist, essentially unique, inner functions Q′, Q′′, of minimal McMillan degree, for which

\{ \begin{matrix}W & = & W_{−}^{e}Q′ \\ W_{+}^{e} & = & W Q′′. \end{matrix}  (55)

The inner functions Q′, Q′′ are uniquely determined by the normalization Q′(∞) = Q′′(∞) = I. We shall refer to the factorization W = W_{−}^{e}Q′ as an outer-inner factorization.

2. Let \overline{W} _{−}^{e} and \overline{W} _{+}^{e} be the extended, antistable, minimum and maximum phase respectively, spectral factors. Given any minimal antistable spectral factor \overline{W}, there exist essentially unique inner functions \overline{Q}′, \overline{Q}′′ for which

\{ \begin{matrix}\overline{W} & = & \overline{W} _{−}^{e}\overline{Q}′ \\ \overline{W} _{+}^{e} & = & \overline{W} \overline{Q}′′. \end{matrix}  (56)

The inner functions \overline{Q}′, \overline{Q}′′ are uniquely determined by the normalization \overline{Q}′(∞) = \overline{Q}′′(∞) = I.

Proof:

1. Let W be a minimal p×m stable spectral factor. Then, by Proposition 3.1, there exists an

essentially unique row rigid m_{0} × m function Q̂′ for which W = W_{−}Q̂′. Let Q′ = ( Q̂′ ; Q̃′ )

be the minimal inner extension of Q̂′. This extension exists by Proposition 5.1 and is

unique up to a constant left unitary factor of the form ( \begin{matrix}I & 0 \\ 0 & V \end{matrix} ). We clearly have

W = W_{−}Q̂′ = ( W_{−} 0 ) ( Q̂′ ; Q̃′ ) = W_{−}^{e}Q′.

By Proposition 5.1, we have δ(Q′) = δ(Q̂′).

Conversely, let W = W_{−}^{e}Q′_{1} with Q′_{1} inner of minimal McMillan degree. Set Q′_{1} = ( Q̂′_{1} ; Q̃′_{1} ).

Then W = W_{−}Q̂′_{1} = W_{−}Q̂′, which by the left invertibility of W_{−} implies Q̂′_{1}(Q̂′_{1})^{∗} = Q̂′(Q̂′)^{∗} = I, it follows by minimality that Q̂′_{1} = V Q̂′ for some inner function V. If δ(Q̂′_{1}) = δ(Q̂′) then this implies that V is necessarily constant.

2. The proof is similar and we omit it.

As a consequence of the previous proposition, we have the following characterization of minimal, stable, spectral factors.

Theorem 5.1 Given a rational spectral function Φ, let W_{+}^{e} be its extended maximum phase, stable spectral factor. Then W ∈ H_{+}^{∞} is a minimal, stable, spectral factor if and only if there exists an inner function Q′′ such that W = W_{+}^{e}(Q′′)^{∗}.

Proof: By Proposition 5.2, if W is a minimal, stable, spectral factor, we have W = W_{+}^{e}(Q′′)^{∗} for some inner function Q′′. Conversely, assume W = W_{+}^{e}(Q′′)^{∗} ∈ H_{+}^{∞} for an inner function Q′′. Clearly, W is a stable spectral factor and it remains to show that it is a minimal factor. Applying Lemma 3.5.7 in Fuhrmann [1981], we have Im Ĥ_{W}^{c} ⊂ Im Ĥ_{W_{+}^{e}}^{c} and hence

δ(W) = dim Im Ĥ_{W}^{c} ≤ dim Im Ĥ_{W_{+}^{e}}^{c} = δ(W_{+}^{e}).

In turn, this implies

2δ(W_{+}^{e}) = δ(Φ) ≤ 2δ(W) ≤ 2δ(W_{+}^{e}).

Thus we must have equality throughout, which proves the minimality of W.

Proposition 5.2 dealt with the zeros of a stable or antistable spectral factor. The next theorem utilizes the DSS factorization to bring in the pole structure of spectral factors and its relation to zeros.

Theorem 5.2 Let W be a minimal, p × m, stable, spectral factor W, and let

W = \overline{W} K = \overline{W} K^{−∗}  (57)

be its right DSS factorization over H_{−}^{∞}. Then

1. \overline{W} is a minimal, antistable spectral factor. Moreover

\overline{W} = W K^{−1} = W K^{∗}  (58)

is a right DSS factorization of \overline{W} over H_{+}^{∞}.

2. Let Q′, Q′′, \overline{Q}′, \overline{Q}′′ be the inner functions whose existence is guaranteed by Proposition 5.2. Then the following diagram is commutative.

[figure: ^{\overline{W} {e \atop −}}; ^{\overline{W} {e \atop +}}; W_{+}^{e}; W_{−}^{e}; W; \overline{W}; Q′; Q′′; ^{\overline{Q}′}; ^{\overline{Q}′′}; Q_{+}^{e}; Q_{−}^{e}; K_{−}^{e}; K; K_{+}^{e}]

Diagram 3

3. The following coprimeness conditions hold. K_{−}^{e} ∧_{L} \overline{Q}′ = I, K ∧_{L} \overline{Q}′′ = I, K ∧_{R} Q′ = I, K_{+}^{e} ∧_{R} Q′′ = I.

Proof:

1. Since K is inner, \overline{W} is clearly an antistable, spectral factor. From (57) we get (58). Since \overline{W} is a spectral factor, we have

δ(W) ≤ δ(\overline{W}) = δ(W K^{−1}) ≤ δ(K^{∗}) = δ(K) = δ(W).

Thus we have equality throughout and \overline{W} is a minimal spectral factor. This also shows the right coprimeness of W, K, i.e. (58) is a right coprime DSS factorization.

2. The upper triangle of diagram 3 follows from (15) and (55). Similarly, the lower triangle follows from (15) and (56). The square is the same as Diagram 2. Finally, (55), (56) and (57) yield W_{+}^{e} = W Q′′ = \overline{W} KQ′′ = \overline{W} _{−}^{e}\overline{Q}′KQ′′ which is the middle part of the diagram.

3. The first relation follows from the fact that from (56) and (58), we can write \overline{W} _{−}^{e} = \overline{W} (\overline{Q}′)^{∗} = W_{−}^{e}(K_{−}^{e})^{∗}. If \overline{Q}′ and K_{−}^{e} were not left coprime, then we could multiply on the right by the inverse of the common factor, obtaining another factorization of W_{−}^{e}; but then (58) would not be a DSS factorization. A similar argument applies to the relation \overline{W} = W_{+}^{e}(\overline{Q}′′)^{∗} = W K^{∗} (derived also from (56) and (58)), to W = W_{−}^{e}Q′ = \overline{W} K and to W_{+}^{e} = W Q′′ = \overline{W} _{+}^{e}K_{+}^{e} (both from (55) and (57))

We will find the following lemma useful in the sequel.

Lemma 5.1 Let W_{1}, W_{2} be two minimal, stable spectral factors having the right coprime DSS factorizations, over H_{−}^{∞}, given by W_{i} = \overline{W}_{i}K_{i}. Assume W_{2} = W_{1}T for some inner function T, and assume T is the inner function with smallest McMillan degree such that the relation holds (in the sense of Proposition 5.1). Then there exists a unique inner function \overline{T} for which K_{1}T = \overline{T} K_{2} and \overline{W}_{2} = \overline{W}_{1}\overline{T}.

Proof: By our assumption we have \overline{W}_{2}K_{2} = W_{2} = W_{1}T = \overline{W}_{1}K_{1}T. Thus we have two right DSS factorizations of W_{2} of which \overline{W}_{2}K_{2} is right coprime. There exists therefore a unique inner function \overline{T} for which \overline{W}_{1} = \overline{W}_{2}\overline{T}^{∗} and K_{1}T = \overline{T} K_{2}. This also implies the equality \overline{W}_{1}\overline{T} = \overline{W}_{2}.

The rigid functions, characterized by Proposition 5.2, can be further factored and this leads to a finer analysis of the set of spectral factors.

Thus we consider next the subsidiary factorizations

\{ \begin{matrix}Q̂′ & = & Q̂′_{1}Q̂′_{2} \\ Q̂′′ & = & Q̂′′_{2}Q̂′′_{1} \end{matrix}  (59)

Here Q̂′_{1}, Q̂′′_{1} are m_{0} × m_{0} inner whereas Q̂′_{2}, Q̂′′_{2} are right and left outer, i.e. are right and left invertible respectively over H_{+}^{∞}. That such factorizations are possible is a direct result of Beurling’s theorem. We will say that Q̂′_{1} and Q̂′′_{1} describe the internal antistable and stable zeros respectively of W, whereas Q̂′_{2} and Q̂′′_{2} describe the external antistable and stable zeros respectively of W. We shall refer to the factorizations (59) as the internal-external and external-internal factorizations of Q̂′ and Q̂′′ respectively. The terminology is fitting as the factorizations in (59) are inner-outer and outer-inner factorizations.

We recall that we need not deal with rigid functions if we utilize the results of Proposition 5.2. Let W be a p × m minimal stable spectral factor. Let W_{−}^{e}, W_{+}^{e} be the appropriately extended, i.e. of dimension p × m, extremal spectral factors. The extension is obtained by adding m − m_{0} zero columns to W_{−} and W_{+} respectively. Let Q′, Q′′ be the inner functions whose existence is guaranteed by Proposition 5.2. Thus

Q′ = ( Q̂′ ; Q̃′ ) , Q′′ = ( Q̂′′ Q̃′′ )

with Q̃′, Q̃′′ given by the embedding procedure described in Proposition 5.1.

The factorizations (59) induce the following factorizations

Q′ = ( \begin{matrix}Q̂′_{1} & 0 \\ 0 & I \end{matrix} ) ( Q̂′_{2} ; Q̃′ ) = Q′_{1}Q′_{2},  (60)

and

Q′′ = ( Q̂′′_{2} Q̃′′ ) ( \begin{matrix}Q̂′′_{1} & 0 \\ 0 & I \end{matrix} ) = Q′′_{2}Q′′_{1}.  (61)

We shall refer to these factorizations as internal-external and external-internal factorizations of Q′ and Q′′ respectively.

The interesting phenomena emerging in the study of rectangular spectral factors is the fact that the internal zeros, namely the zeros parametrized by the inner functions Q′_{1} and Q′′_{1} no longer account for all the (stable) zeros of W_{−} or the (antistable) zeros of W_{+}. These are measured by the inner function Q_{+}. So some zeros have been externalized. This is a central theme in the parametrization of the set of stable spectral factors. The extreme situation is encountered in the set of external factors, a class that will be introduced in Definition 5.2.

The following theorem analyzes the situation in the general case, and brings in a new inner function. It will be seen later that this inner function is a measure of the number of external zeros of the spectral factor. A geometric interpretation of this inner function in terms of reachability subspaces will be given in a subsequent paper.

We turn our attention now to the study of arbitrary, minimal p × m spectral factors. Essentially, the next theorem provides a parametrization of the set of all p × m, minimal, stable spectral factors. This will be made explicit in Theorem 5.6.

Theorem 5.3 Let Φ be a p × p, rational spectral function of rank m_{0}. Let W be a minimal, p × m, stable spectral factor. Let W = \overline{W} K be the DSS factorization over H_{−}^{∞}. Let Q_{+}, Q_{−} be the m_{0} × m_{0} inner functions for which

\{ \begin{matrix}W_{−}Q_{+} & = & W_{+} \\ \overline{W}_{−}Q_{−} & = & \overline{W}_{+}. \end{matrix}  (62)

Let Q′, Q′′, \overline{Q}′, \overline{Q}′′ be determined by Proposition 5.2 have the internal-external factorizations

Q′ = Q′_{1}Q′_{2},  \overline{Q}′ = \overline{Q}′_{1}\overline{Q}′_{2},  (63)

and the external-internal factorizations

Q′′ = Q′′_{2}Q′′_{1},  \overline{Q}′′ = \overline{Q}′′_{2}\overline{Q}′′_{1}.  (64)

Then 1. (a) We have Q′_{1} is the greatest common left inner factor of Q′ and Q_{+}^{e} and Q′′_{1} is the

greatest common right inner factor of Q′′ and Q_{+}^{e}. We have \overline{Q}′_{1} is the greatest common left inner factor of \overline{Q}′ and \overline{Q}_{+}^{e} and \overline{Q}′′_{1} is the greatest common right inner factor of \overline{Q}′′ and \overline{Q}_{+}^{e}.

(b) \overline{Q}′′_{1} is the greatest common right inner factor of \overline{Q}′′ and \overline{Q}_{−}^{e} and Q′′_{1} is the greatest common right inner factor of Q′′ and Q_{+}^{e}.

2. We have

Q′Q′′ = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )  (65)

with R inner. We have

\overline{Q}′\overline{Q}′′ = ( \begin{matrix}Q_{−} & 0 \\ 0 & \overline{R} \end{matrix} )  (66)

with \overline{R} inner.

3. There exists a unique inner function Q such that

Q_{+}^{e} = Q′_{1}QQ′′_{1}.  (67)

A similar factorization holds for \overline{Q}_{+}^{e}, namely

\overline{Q}_{+}^{e} = \overline{Q}′_{1}\overline{Q}\overline{Q}′′_{1}.  (68)

4. We have the following equalities

\{ \begin{matrix}K_{−}^{e}Q′ & = & \overline{Q}′K \\ \overline{Q}′′K_{+}^{e} & = & KQ′′ \end{matrix}  (69)

Here K_{−}^{e}, \overline{Q}′ are left coprime and Q′, K right coprime. Similarly, K, \overline{Q}′′ are left coprime and K_{+}^{e}, Q′′ right coprime.

5. Define, after Lindquist and Picci [1991],

\{ \begin{matrix}W_{0−} & = & W_{−}^{e}Q′_{1} \\ W_{0+} & = & W_{+}^{e}(Q′′_{1})^{∗} \end{matrix}  (70)

Then we have

\{ \begin{matrix}W & = & W_{0−}Q′_{2} \\ W Q′′_{2} & = & W_{0+} \end{matrix}  (71)

6. Let

\{ \begin{matrix}W_{0−} & = & \overline{W}_{0−}K_{0−} \\ W_{0+} & = & \overline{W}_{0+}K_{0+} \end{matrix}  (72)

be right coprime DSS factorizations over H_{−}^{∞} . Then

\{ \begin{matrix}K_{−}^{e}Q′_{1} & = & \overline{Q}′_{1}K_{0−} \\ \overline{Q}′′_{1}K_{+}^{e} & = & K_{0+}Q′′_{1} \end{matrix}  (73)

and

K_{0−}Q = \overline{Q}K_{0+}.  (74)

7. We have

\overline{Q}′KQ′′ = ( \begin{matrix}Q_{−}K_{+} & 0 \\ 0 & R_{W} \end{matrix} )  (75)

with R_{W} inner, and moreover

R = \overline{R} = R_{W} .  (76)

8. (a) The inner functions Q and Q′_{2} are left coprime and Q and Q′′_{2} are right coprime. (b) The inner functions \overline{Q} and \overline{Q}′_{2} are left coprime and \overline{Q} and \overline{Q}′′_{2} are right coprime.

(c) The inner functions Q′_{2}, R^{e} are left coprime and Q′′_{2}, R^{e} are right coprime.

(d) The inner functions \overline{Q}′_{2}, R^{e} are left coprime and \overline{Q}′′_{2}, R^{e} are right coprime.

9. The inner functions Q, \overline{Q}, Q′_{2}, Q′′_{2}, \overline{Q}′_{2}, \overline{Q}′′_{2}, R are all equivalent.

10. The inner functions K_{−}^{e}, K_{0−}, K, K_{0+}, K_{+}^{e} are all equivalent.

11. The following diagram is commutative.

[figure: ^{\overline{W} {e \atop −}}; ^{\overline{W} {e \atop +}}; W_{+}^{e}; W_{−}^{e}; W; \overline{W}; W_{0−}; W_{0+}; ^{\overline{W}^{0−}}; ^{\overline{W}^{0+}}; Q′; Q′′; ^{\overline{Q}′}; ^{\overline{Q}′′}; Q′_{1}; ^{\overline{Q}{′ \atop 1}}; Q′′_{1}; ^{\overline{Q}{′′ \atop 1}}; Q′′_{2}; ^{\overline{Q}{′′ \atop 2}}; Q′_{2}; ^{\overline{Q}{′ \atop 2}}; Q; \overline{Q}; K_{−}^{e}; K; K_{+}^{e}; K_{0−}; K_{0+}]

Diagram 4

Proof:

1. (a) Recall that in the factorization Q̂′ = Q̂′_{1}Q̂′_{2} the factor Q̂′_{2} is outer. Since Q̂′Q̂′′ = Q_{+}, Q̂′_{1} is a left inner factor of Q_{+}. This implies that Q′ = ( \begin{matrix}Q̂′_{1} & 0 \\ 0 & I \end{matrix} ) is a common left inner factor of Q′, Q_{+}^{e}. Any left inner factor of Q_{+}^{e} is, up to a constant right unitary factor, of the form ( \begin{matrix}Q̂ & 0 \\ 0 & I \end{matrix} ). Since Q̂′_{2} is outer, it is clear that Q′_{1} is the greatest common left inner factor of Q′ and Q_{+}^{e}. The second assertion is proved analogously.

(b) The second part follows by duality considerations, working over H_{−}^{∞} and starting the analysis from \overline{W}_{+}.

2. Note that from

Q′ = ( Q̂′ ; Q̃′ ) , Q′′ = ( Q̂′′ Q̃′′ )

we obtain

\begin{aligned}Q′Q′′ & = & ( Q̂′ ; Q̃′ ) ( Q̂′′ Q̃′′ ) = ( \begin{matrix}Q̂′Q̂′′ & Q̂′Q̃′′ \\ Q̃′Q̂′′ & Q̃′Q̃′′ \end{matrix} ) \\ & = & ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) . \end{aligned}

Here we used the fact that W_{−}Q̂′Q̂′′ = W_{+} and hence Q̂′Q̂′′ = Q_{+}. Since Q′Q′′ is inner, it follows, necessarily, that Q̂′Q̃′′ = 0 and Q̃′Q̂′′ = 0. Finally, this also implies that R = Q̃′Q̃′′ is inner.

3. Note that Q̂′Q̃′′ = 0 implies Q̂′_{2}Q̃′′ = 0 and Q̃′Q̂′′ = 0 implies Q̃′Q̂′′_{2} = 0. Using this, we compute

W_{+}^{e} = W_{−}^{e}Q′Q′′ = W_{−}^{e} ( \begin{matrix}Q̂′_{1} & 0 \\ 0 & I \end{matrix} ) ( Q̂′_{2} ; Q̃′ ) ( Q̂′′_{2} Q̃′′ ) ( \begin{matrix}Q̂′′_{1} & 0 \\ 0 & I \end{matrix} )
= W_{−}^{e} ( \begin{matrix}Q̂′_{1} & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}Q̂′_{2}Q̂′′_{2} & Q̂′_{2}Q̃′′ \\ Q̃′Q̂′′_{2} & Q̃′Q̃′′ \end{matrix} ) ( \begin{matrix}Q̂′′_{1} & 0 \\ 0 & I \end{matrix} )
= W_{−}^{e} ( \begin{matrix}Q̂′_{1} & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}Q̂ & 0 \\ 0 & R \end{matrix} ) ( \begin{matrix}Q̂′′_{1} & 0 \\ 0 & I \end{matrix} )

But we also have

W_{+}^{e} = W_{−}^{e} ( \begin{matrix}Q_{+} & 0 \\ 0 & I \end{matrix} )

This implies the equality Q_{+} = Q̂′_{1}Q̂Q̂′′_{1}.

4. Since W is a minimal, stable spectral factor, we have W = W_{−}Q̂′ = \overline{W}_{−}K_{−}Q̂′. But we also have W = \overline{W} K = \overline{W}_{−} ˆ\overline{Q}′K. Using the left invertibility of \overline{W}_{−} and comparing the two expressions, we have

K_{−}Q̂′ = ˆ \overline{Q}′K.  (77)

Since K is the minimal inner function that stabilizes \overline{W}, then necessarily Q̂′ and K are right coprime. Clearly, this implies the right coprimeness of Q′ and K. Considering the respective Bezout equations, this shows also the right coprimeness of Q′ and K. Similarly, K_{−}^{∗} is the minimal conjugate inner function that destabilizes W_{−}, which implies the left

coprimeness of K_{−} and ˆ \overline{Q}′. Equation (77), together with the coprimeness conditions, shows that K_{−} and K are equivalent inner functions. In particular, they have the same McMillan degree. Also it follows that the McMillan degrees of Q̂′, ˆ \overline{Q}′ are equal. Now

we look at Q′ = ( Q̂′ ; Q̃′ ) and ( \begin{matrix}ˆ\overline{Q} & ′ \\ ˜\overline{Q} & ′ \end{matrix} ) which are both minimal McMillan degree inner completions. So δ(Q′) = δ(Q̂′) = δ( ˆ\overline{Q}′) = δ(\overline{Q}′). This shows that \overline{Q}′K = ( \begin{matrix}ˆ\overline{Q}′K \\ ˜\overline{Q}′K \end{matrix} ) =

( \begin{matrix}K_{−}Q̂′ \\ ˜\overline{Q}′K \end{matrix} ) is a minimal inner extension of ˆ \overline{Q}′K. Since a minimal inner extension is unique

up to a left constant inner factor which we absorb into ˆ \overline{Q}′, we conclude that Q̃′ = ˜ \overline{Q}′K. Thus the first equality in (69) follows.

The second equality follows by duality considerations, working over H_{−}^{∞} and starting the analysis from \overline{W}_{+}.

5. We compute W = W_{−}^{e}Q′ = W_{−}^{e}Q′_{1}Q′_{2} = W_{0−}Q′_{2}. Similarly, W_{+}^{e} = W Q′′ = W Q′′_{2}Q′′_{1}, and so W_{0+} = W_{+}^{e}(Q′′_{1})^{∗} = W Q′′_{2}.

6. The equalities in (73) are a special case of those in (69), replacing W by W_{0−} for the first one and by W_{0+} for the second. The existence of \overline{Q} follows also from that of Q by an application of Lemma 5.1.

7. Using the equalities (69), we have

K_{−}^{e}Q′Q′′ = \overline{Q}′KQ′′ = \overline{Q}′\overline{Q}′′K_{+}^{e}.

Substituting equalities (65) and (66), we get

\begin{aligned}\overline{Q}′KQ′′ & = & K_{−}^{e}Q′Q′′ = K_{−}^{e} ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) = ( \begin{matrix}K_{−}Q_{+} & 0 \\ 0 & R \end{matrix} ) \\ & = & \overline{Q}′\overline{Q}′′K_{+}^{e} = ( \begin{matrix}Q_{−} & 0 \\ 0 & \overline{R} \end{matrix} ) K_{+}^{e} = ( \begin{matrix}Q_{−}K_{+} & 0 \\ 0 & \overline{R} \end{matrix} ) \end{aligned}

Since we have the equality K_{−}Q_{+} = Q_{−}K_{+}, it follows that R = \overline{R} and that

\overline{Q}′KQ′′ = ( \begin{matrix}Q_{−}K_{+} & 0 \\ 0 & \overline{R} \end{matrix} ) .

8. (a) Since Q′_{1} is the greatest common left inner factor of Q′ and Q_{+}^{e}, it follows that Q′_{2}, QQ′′_{1} are left coprime and so are Q′_{2}, Q. In a similar way, the right coprimeness of the pair Q′′_{2}, Q is proved.

(b) In much the same way, the left coprimeness of \overline{Q}′_{2}, \overline{Q} and the right coprimeness of \overline{Q}′′_{2}, \overline{Q} is also proved.

(c) A common left inner factor of Q′_{2} = ( Q̂′ ; Q̃′ ) and ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ) is, up to a right constant

unitary factor, of the form ( \begin{matrix}I & 0 \\ 0 & P \end{matrix} ). If P were nontrivial, it would contradict the

assumption that Q′ is the minimal McMillan degree extension of Q̂′. Hence the right coprimeness.

(d) The proof is similar.

9. We use the equalities

\begin{aligned}Q′_{2}Q′′_{2} & = & ( \begin{matrix}Q̂ & 0 \\ 0 & R \end{matrix} ) = ( \begin{matrix}Q̂ & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ) ( \begin{matrix}Q̂ & 0 \\ 0 & I \end{matrix} ) \\ \overline{Q}′_{2}\overline{Q}′′_{2} & = & ( \begin{matrix}ˆ\overline{Q} & 0 \\ 0 & R \end{matrix} ) = ( \begin{matrix}ˆ\overline{Q} & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ) ( \begin{matrix}ˆ\overline{Q} & 0 \\ 0 & I \end{matrix} ) \end{aligned}  (78)

These equalities, together with the proven coprimeness conditions, imply now the equivalences

\begin{matrix}Q′_{2} ≃ ( \begin{matrix}Q̂ & 0 \\ 0 & I \end{matrix} ) ≃ ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ) ≃ Q ≃ R \\ \overline{Q}′_{2} ≃ ( \begin{matrix}ˆ\overline{Q} & 0 \\ 0 & I \end{matrix} ) ≃ ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ) ≃ \overline{Q} ≃ R. \end{matrix}

By transitivity of the equivalence of inner functions, the statement is proved.

10. The equality KQ′′ = \overline{Q}′′K_{+}^{e}, the right coprimeness of Q′′, K_{+}^{e} and the left coprimeness of K, \overline{Q}′′ imply the equivalence of K and K_{+}^{e}. The rest follows by the transitivity of equivalence.

11. Equalities (63), (64), (69), (73), (70), (71), (74) taken together, yield the commutativity of the diagram.

Definition 5.1 We denote by π_{1} the projection of C^{m} on the subspace of all vectors whose last m − m_{0} coordinates vanish, and by π_{2} its complement, i.e. π_{2} := I − π_{1}

Corollary 5.1 Given the factorizations (65) and (66) with Q_{±} both m_{0} × m_{0} inner functions, then we have

\{ \begin{matrix}H_{r}(Q′′)π_{1} & ⊂ & H_{r}(Q_{+}) \\ H_{r}(\overline{Q}′′)π_{1} & ⊂ & H_{r}(Q_{−}) \end{matrix}  (79)

Proof: The factorization (65) can be written as Q′Q′′ = Q_{+}^{e}R^{e} = R^{e}Q_{+}^{e} and therefore we have

H_{r}(Q′Q′′) = H_{r}(Q′)Q′′ ⊕ H_{r}(Q′′) = H_{r}(R^{e}) ⊕ H_{r}(Q_{+}^{e}),

and hence the inclusion H_{r}(Q′′) ⊂ H_{r}(R^{e}) ⊕ H_{r}(Q_{+}^{e}). Applying the projection π_{1} to this inclusion, we get H_{r}(Q′′)π_{1} ⊂ H_{r}(Q_{+}). The other inclusion is proved similarly.

Theorem 5.3 is general but two special cases need to be pointed out. This leads us to the following definition.

Definition 5.2 Let W be a p × m minimal, stable spectral factor and let Q′, Q′′ be the minimal inner functions characterized in Proposition 5.2.

1. We say W is an internal spectral factor if we have Q′Q′′ = Q_{+}^{e}.

2. We say W is an external spectral factor if we have

(a) Q′, Q_{+}^{e} are left coprime, and

(b) Q′′, Q_{+}^{e} are right coprime.

We point out that our definition of external spectral factors differs from the way Lindquist and Picci use this term. In their work they equate external with noninternal.

A description of factorizations of inner functions is clearly needed. A complete characterization of all factorizations of inner functions is available in terms of nonnegative definite solutions of a homogeneous Riccati equation or in terms of invariant subspaces of a linear transformation. For more on this, see Willems [1972], Finesso and Picci [1982], Picci and Pinzoni [1993] and Fuhrmann [1995].

External and internal spectral factors are obviously defined in an extremely opposite way. The analysis of the general spectral factors will depend to a large extent on a full understanding of the classes of both external and internal spectral factors.

The next proposition is a characterization of minimal, stable, internal spectral factors.

Proposition 5.3 Let W be a p × m minimal, stable spectral factor. Let Q′, Q′′ be the inner functions determined via Proposition 5.2. Then W is an internal spectral factor if and only if we have

Q′Q′′ = Q_{+}^{e} = ( \begin{matrix}Q_{+} & 0 \\ 0 & I \end{matrix} ) .  (80)

and, up to a right unitary factor for Q′ and the inverse left factor for Q′′, the factorizations (80) are in a bijective correspondence with normalized inner factorizations of Q_{+}, that is factorizations Q_{+} = Q_{1}Q_{2} with Q_{i} normalized inner functions.

Proof: From W = W_{−}^{e}Q′ and the fact that the last m − m_{0} columns of both W and W_{−}^{e} are

zero, it follows that Q′ = ( \begin{matrix}Q_{1} & 0 \\ 0 & I \end{matrix} ) for some m_{0} × m_{0} inner function Q_{1}. A similar argument

holds for Q′′. Clearly, we have Q_{1}Q_{2} = Q_{+}.

Corollary 5.2 Let W be a p × m_{0} minimal, stable spectral factor. Then W is necessarily internal.

Proof: As in the proof of Proposition 5.3, we have W = W_{−}^{e}Q′ with Q′ = ( \begin{matrix}Q_{1} & 0 \\ 0 & I \end{matrix} ) for some

m_{0} × m_{0} inner function Q_{1}. Necessarily Q_{1} is a left factor of Q_{+}, i.e. W is internal.

From the proof of Theorem 5.3 it becomes clear that minimal, stable, external spectral factors of size p × m of a given spectral density Φ are related to factorizations of an inner function of the form ( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} ), having special coprimeness properties. We proceed to formalize

these properties in the following definition.

Definition 5.3 Let Q be an m_{0} × m_{0} inner function and let R be an (m − m_{0}) × (m − m_{0}) inner function equivalent to Q. We say that a factorization

Q′Q′′ = ( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} )  (81)

into the product of two m×m inner functions is a balanced factorization if Q′ is left coprime

and Q′′ right coprime with both Q^{e} = ( \begin{matrix}Q & 0 \\ 0 & I \end{matrix} ) and R^{e} = ( \begin{matrix}I & 0 \\ 0 & R \end{matrix} ).

Note that for the factorization (81) to be balanced, we need four coprimeness conditions. However, two of them are trivially satisfied. In fact, by the construction in Theorem 5.2, we have the left coprimeness of Q′, R^{e} as well as the right coprimeness of Q′′, R^{e}.

In the following we will need the next lemma.

Lemma 5.2 Let

Q′Q′′ = ( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} ) .  (82)

Let, given an inner function Q, the unitary map τQ̃ : H_{r}(Q) ⟶ H_{r}(Q̃) be defined by

(fτQ̃)(s) = f(−s)Q̃(s)  (83)

Here Q̃(s) = Q(\overline{s})^{∗}. Then

1. The following diagrams are commutative

[figure: H_{r}(Q_{+}); H_{r}(Q̃_{+}); H_{r}(Q̃′′Q̃′); H_{r}(Q′Q′′); τ_{Q̃′′Q̃′}; τQ̃_{+}^{e}; π_{1}; π_{1}]

Diagram 5 and

[figure: H_{r}(R); H_{r}(R̃); H_{r}(Q̃′′Q̃′); H_{r}(Q′Q′′); τ_{Q̃′′Q̃′}; τR̃; π_{2}; π_{2}]

Diagram 6

2. We have

\{ \begin{matrix}H_{r}(Q′′)τ_{Q̃′′Q̃′} & = & H_{r}(Q̃′′)Q̃′ \\ H_{r}(Q′)Q′′τ_{Q̃′′Q̃′} & = & H_{r}(Q̃′) \end{matrix}  (84)

Proof:

1. From the factorization (82) it follows that

τ_{Q̃′′Q̃′} = τ_{( \begin{matrix}Q̃_{+} & 0 \\ 0 & R̃ \end{matrix} )} = τQ̃_{+}^{e} ⊕ τR̃^{e}.

Moreover, since H_{r}(Q′Q′′) = H_{r}(Q_{+}^{e}) ⊕ H_{r}(R^{e}), any f ∈ H_{r}(Q′Q′′) can be written as f = ( f_{1} f_{2} ) with f_{1} ∈ H_{r}(Q_{+}^{e}) and f_{2} ∈ H_{r}(R^{e}). With this we compute

fτ_{Q̃′′Q̃′}π_{1} = ( f_{1} f_{2} ) (τQ̃_{+}^{e} ⊕ τR̃^{e})π_{1}
= ( f_{1}τQ̃_{+}^{e} f_{2}τR̃^{e} ) π_{1}
= f_{1}τQ̃_{+}^{e} = fπ_{1}τQ̃_{+}^{e}.

The commutativity of the second diagram is proved similarly.

2. Let f ∈ H_{r}(Q′′). Then fτ_{Q̃′′Q̃′} = (fτQ̃′′)Q̃′. Since H_{r}(Q′′)τQ̃′′ = H_{r}(Q̃′′), the first equality is proved. The proof of the second equality is similar.

The following proposition is a geometric characterization of balanced factorizations. It is advisable, in following the arguments, to refer to Diagram (4).

5 RECTANGULAR SPECTRAL FACTORS 35 Proposition 5.4 Let W be a minimal, stable spectral factor. Then, with the notation of Theorem 5.3, we have

1. The factorization

Q′_{2}Q′′_{2} = ( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} )  (85)

is a balanced factorization.

2. We have

\{ \begin{matrix}H_{r}(Q′′_{2})π_{1} = H_{r}(Q) \\ H_{r}(Q′′_{2})π_{2} = H_{r}(R) \end{matrix}  (86)

3. We have

\{ \begin{matrix}H_{r}(\overline{Q}′′_{2})π_{1} = H_{r}(\overline{Q}) \\ H_{r}(\overline{Q}′′_{2})π_{2} = H_{r}(R) \end{matrix}  (87)

4. We have

\{ \begin{matrix}π_{1}H_{c}(Q′_{2}) = H_{c}(Q^{e}) \\ π_{2}H_{c}(Q′_{2}) = H_{c}(R^{e}) \end{matrix}  (88)

5. We have

\{ \begin{matrix}H_{r}(Q′′)π_{1} = H_{r}(QQ′′_{1}) \\ H_{r}(Q′)Q′′π_{1} = H_{r}(Q′_{1}Q)Q′′_{1}. \end{matrix}  (89)

6. We have

\{ \begin{matrix}H_{r}(Q′′)π_{2} = H_{r}(R) \\ H_{r}(Q′)Q′′π_{2} = H_{r}(R). \end{matrix}  (90)

7. We have

\{ \begin{matrix}H_{r}(\overline{Q}′′)π_{2} = H_{r}(R) \\ H_{r}(\overline{Q}′)\overline{Q}′′π_{2} = H_{r}(R). \end{matrix}  (91)

8. We have

H_{r}(QQ′′_{1}) ∩ H_{r}(Q′_{1}Q)Q′′_{1} = H_{r}(Q)Q′′_{1},  (92)

or equivalently

H_{r}(Q′′)π_{1} ∩ H_{r}(Q′)Q′′π_{1} = H_{r}(Q)Q′′_{1}.

9. The reduced maps H_{r}(Q′′)|π_{1} and H_{r}(Q′)Q′′|π_{1} are injective.

10. The reduced maps H_{r}(Q′′)|P_{H_{r}(Q_{+}^{e})} and H_{r}(Q′)Q′′|P_{H_{r}(Q_{+}^{e})} are injective, and moreover, we

have

\begin{aligned}H_{r}(Q′′)P_{H_{r}(Q_{+}^{e})} & = & H_{r}(QQ′′_{1}) \\ H_{r}(Q′)Q′′P_{H_{r}(Q_{+}^{e})} & = & H_{r}(Q′_{1}Q)Q′′_{1} \end{aligned}  (93)

5 RECTANGULAR SPECTRAL FACTORS 36

11. We have

H_{r}(Q_{+}^{e})P_{H_{r}(Q′′)} = H_{r}(Q′′),  (94)

i.e. H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′)} is surjective.

Proof:

1. That the factorization (85) is balanced was proved in Theorem 5.3.

2. From the factorization (85) it follows that

H_{r}(Q′′_{2}) ⊂ H_{r}(Q′_{2}Q′′_{2}) = H_{r}(Q^{e}) ⊕ H_{r}(R^{e}),  (95)

and hence

H_{r}(Q′′_{2})π_{1} ⊂ H_{r}(Q′_{2}Q′′_{2}) = H_{r}(Q^{e}).

From the fact that the factorization is balanced, we conclude the equivalence of the inner functions Q′′_{2} and Q, and hence it suffices to show that Ker H_{r}(Q′′_{2})|π_{1} = {0}. Indeed, assume f = ( f_{1} f_{2} ) ∈ H_{r}(Q′′_{2}) and fπ_{1} = 0, i.e. f_{1} = 0. Thus, from (95), necessarily ( 0 f_{2} ) ∈ H_{r}(R^{e}). So f ∈ H_{r}(Q′′_{2}) ∩ H_{r}(R^{e}). But this intersection is zero by the right coprimeness of Q′′_{2} and R^{e}.

3. This is proved in a completely analogous way to the proof of the previous assertion.

4. This is proved in a completely analogous way to the proof of the previous assertion.

5. From Q′′ = Q′′_{2}Q′′_{1} it follows that H_{r}(Q′′) = H_{r}(Q′′_{2})Q′′_{1} ⊕ H_{r}(Q′′_{1}). We apply now π_{1} to this equality, noting that π_{1} commutes with multiplication by Q′′_{1} and that H_{r}(Q′′_{1})π_{1} = H_{r}(Q′′_{1}). Hence

H_{r}(Q′′)π_{1} = H_{r}(Q′′_{2})Q′′_{1}π_{1} + H_{r}(Q′′_{1})
= H_{r}(Q′′_{2})π_{1}Q′′_{1} ⊕ H_{r}(Q′′_{1})
= H_{r}(Q)Q′′_{1} ⊕ H_{r}(Q′′_{1}) = H_{r}(QQ′′_{1}).

Here we used the previously established equality H_{r}(Q′′_{2})π_{1} = H_{r}(Q). To prove the second equality in (89), we apply the unitary map τQ̃_{+} to the first, using the commutativity of Diagram 5. We compute

\begin{aligned}H_{r}(Q̃′′_{1}Q̃)Q̃′_{1} & = & H_{r}(QQ′′_{1})τQ̃_{+} = H_{r}(Q′′)π_{1}τQ̃_{+} \\ & = & H_{r}(Q′′)τ_{Q̃′′Q̃′}π_{1} = H_{r}(Q̃′′)Q̃′π_{1}. \end{aligned}

This is equivalent to the statement, using duality and starting from the factorization

Q̃′′Q̃′ = ( \begin{matrix}Q̃_{+} & 0 \\ 0 & R̃ \end{matrix} ).

6. As before, from Q′′ = Q′′_{2}Q′′_{1} it follows that H_{r}(Q′′) = H_{r}(Q′′_{2})Q′′_{1} ⊕ H_{r}(Q′′_{1}). To this

equality we apply now the projection π_{2} and use the second equality in (86) to get H_{r}(Q′′)π_{2} = H_{r}(Q′′_{2})Q′′_{1}π_{2} = H_{r}(Q′′_{2})π_{2} = H_{r}(R).

Using Lemma 5.2, we apply to the previous equality the unitary map τR̃ to get

\begin{aligned}H_{r}(R̃) & = & H_{r}(R)τR̃ = H_{r}(Q′′)π_{2}τR̃ \\ & = & H_{r}(Q′′)τ_{Q̃′′Q̃′}π_{2} = H_{r}(Q̃′′)Q̃′π_{2}. \end{aligned}

This is equivalent to the second equality.

7. This is proved analogously.

8. Follows directly from the equalities in (89). In fact

\begin{aligned}H_{r}(QQ′′_{1}) ∩ H_{r}(Q′_{1}Q)Q′′_{1} & = & [H_{r}(Q)Q′′_{2} ⊕ H_{r}(Q′′_{2})] ∩ [H_{r}(Q′_{1})QQ′′_{2} ⊕ H_{r}(Q)Q′′_{2}] \\ & = & H_{r}(Q)Q′′_{2}, \end{aligned}

as clearly H_{r}(Q′′_{2}) ⊥ H_{r}(Q′_{1})QQ′′_{2}.

9. Using (89), it suffices to show that dim H_{r}(Q′′) = dim H_{r}(QQ′′_{1}). Now Q′′ = Q′′_{2}Q′′_{1} and so H_{r}(Q′′) = H_{r}(Q′′_{2})Q′′_{1} ⊕ H_{r}(Q′′_{1}), which implies the equality dim H_{r}(Q′′) = deg det Q′′_{2} + deg det Q′′_{1}. Now, the fact that the factorization (85) is balanced implies the equality deg det Q′′_{2} = deg det Q. Since dim H_{r}(QQ′′_{1}) = deg det Q + deg det Q′′_{1} the result follows.

10. Clearly, H_{r}(Q_{+}) ⊂ H_{+}^{2}π_{1} and therefore P_{H_{r}(Q_{+})} = π_{1}P_{H_{r}(Q_{+})}. Applying this to the equality H_{r}(Q′′)π_{1} = H_{r}(QQ′′_{1}), and noting that H_{r}(QQ′′_{1}) ⊂ H_{r}(Q_{+}), we get

\begin{aligned}H_{r}(Q′′)P_{H_{r}(Q_{+})} & = & H_{r}(Q′′)π_{1}P_{H_{r}(Q_{+})} \\ & = & H_{r}(QQ′′_{1})P_{H_{r}(Q_{+})} = H_{r}(QQ′′_{1}). \end{aligned}

As we saw that dim H_{r}(Q′′) = dim H_{r}(QQ′′_{1}), the statement is proved.

11. Given a Hilbert space H with subspaces U, V, let P_{U}, P_{V} be the respective orthogonal projections. Then a simple computation yields

(P_{V} |U)^{∗} = P_{U}|V.  (96)

In particular, if P_{V} |U is injective, P_{U}|V is surjective. Thus the statement follows from the preceding one.

We wish to emphasize one point which may be confusing exactly to those readers that have some intuition in the functional approach to geometric control theory. Given two arbitrary inner functions Q′′ and Q_{+}^{e} of the same size, then, in general, H_{r}(Q′′)P_{H_{r}(Q_{+}^{e})} is not a coinvariant subspace of H_{r}(Q_{+}^{e}), but, as will be shown in a forthcoming paper, a controlled invariant, inner stabilizable subspace with respect to a natural controllable pair associated with Q_{+}^{e}. Equation (93) shows that under our assumptions and the construction of Q′, Q′′ in Proposition 5.2, this subspace is actually coinvariant. This observation is crucial for the later analysis of the partial ordering of the set of minimal, stable spectral factors. Also, we point out that the question of existence of balanced facorizations will be addressed in section 7.

5 RECTANGULAR SPECTRAL FACTORS 38

Corollary 5.3 With the notation of Proposition 5.4, we have

1.

\{ \begin{matrix}H_{r}(Q′′)π_{1} & = & H_{r}(Q′′)P_{H_{r}(Q_{+}^{e})} = H_{r}(QQ′′_{1}) \\ H_{r}(Q′)Q′′π_{1} & = & H_{r}(Q′)Q′′P_{H_{r}(Q_{+}^{e})} = H_{r}(Q′_{1}Q)Q′′_{1} \end{matrix}  (97)

and

\{ \begin{matrix}H_{r}(Q′′)π_{2} & = & H_{r}(Q′′)P_{H_{r}(R)} = H_{r}(R) \\ H_{r}(Q′)Q′′π_{2} & = & H_{r}(Q′)Q′′P_{H_{r}(R)} = H_{r}(R) \end{matrix}  (98)

2.

\{ \begin{matrix}H_{r}(\overline{Q}′′)π_{1} & = & H_{r}(\overline{Q}′′)P_{H_{r}(Q_{−}^{e})} = H_{r}(\overline{Q}\overline{Q}′′_{1}) \\ H_{r}(\overline{Q}′)\overline{Q}′′π_{1} & = & H_{r}(\overline{Q}′)\overline{Q}′′P_{H_{r}(Q_{−}^{e})} = H_{r}(\overline{Q}′_{1}\overline{Q})\overline{Q}′′_{1} \end{matrix}  (99)

and

\{ \begin{matrix}H_{r}(\overline{Q}′′)π_{2} & = & H_{r}(\overline{Q}′′)P_{H_{r}(R)} = H_{r}(R) \\ H_{r}(\overline{Q}′)\overline{Q}′′π_{2} & = & H_{r}(\overline{Q}′)\overline{Q}′′P_{H_{r}(R)} = H_{r}(R) \end{matrix}  (100)

Proof: (97) follows from equations (89) and (93).

To prove (98) we use the equalities (90). Note that the inclusion H_{r}(R) ⊂ H_{+}^{2}π_{2} implies fπ_{2}P_{H_{r}(R)} = fP_{H_{r}(R)} for all f ∈ H_{+}^{2}. Therefore

H_{r}(Q′)Q′′P_{H_{r}(R)} = H_{r}(Q′)Q′′π_{2}P_{H_{r}(R)} = H_{r}(R)P_{H_{r}(R)} = H_{r}(R),

and similarly

H_{r}(Q′′)P_{H_{r}(R)} = H_{r}(Q′′)π_{2}P_{H_{r}(R)} = H_{r}(R)P_{H_{r}(R)} = H_{r}(R).

Corollary 5.4 With the notation of Proposition 5.4, we have 1. H_{r}(Q′′)π_{1} = H_{r}(Q_{+}^{e}) if and only if Q′, Q_{+}^{e} are left coprime.

2. H_{r}(Q′)Q′′π_{1} = H_{r}(Q_{+}^{e}) if and only if Q′′, Q_{+}^{e} are right coprime.

3. Both equalities hold if and only if W is external.

We saw that, given a minimal, stable spectral factor W, the inner functions Q′_{1} and Q′′_{1} parametrize the number of internal antistable and stable zeros of W respectively. Thus the previous corollary can be restated in these terms. Corollary 5.5 Let W be as in Proposition 5.4. Then

1. H_{r}(Q′′)π_{1} = H_{r}(Q_{+}^{e}) if and only if W has no internal antistable zeros.

2. H_{r}(Q′′)π_{2} = H_{r}(R^{e}) if and only if W has no internal stable zeros.

3. W is an external spectral factor if and only if the factorization (65) is balanced which in turn is equivalent to the conditions

\{ \begin{matrix}H_{r}(Q′′)π_{1} = H_{r}(Q_{+}^{e}) \\ H_{r}(Q′′)π_{2} = H_{r}(R^{e}) \end{matrix}

The previous results dealt with the geometry of coinvariant and semiinvariant subspaces related to the zeros of a spectral factor W, i.e. to the inner functions appearing in Theorem 5.3. The situation gets more intricate as soon as we add to this the pole structure. This means that we also consider the DSS factorizatios of the spectral factor W, that is we study also spaces associated with the inner function K. Of particular interest is, using the notation of Theorem 5.3, the space H_{r}(K)Q′′ which we associate with the spectral factor W. The full study of these spaces will be given in Section 6. The following results, which reflect some of the results of Corollary 5.3 will be used in later sections.

Lemma 5.3 With the notation of Theorem 5.3, and assuming the factorizations Q′Q′′ =

( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) and \overline{Q}′\overline{Q}′′ = ( \begin{matrix}Q_{−} & 0 \\ 0 & R \end{matrix} ) are balanced, we have

1. The restricted projection map H_{r}(Q′)Q′′|P_{H_{r}(K)Q′′} is injective. Equivalently

H_{r}(K)Q′′P_{H_{r}(Q′)Q′′} = H_{r}(Q′)Q′′.  (101)

2. The restricted projection map H_{r}(R^{e})|P_{H_{r}(K)Q′′} is injective. Equivalently

H_{r}(K)Q′′P_{H_{r}(R^{e})} = H_{r}(R^{e}).  (102)

3. We have

H_{r}(K)Q′′π_{2} = H_{r}(R^{e}).  (103)

4. The restricted projection map H_{r}(\overline{Q}′′)K_{+}^{e}|P_{H_{r}(K)Q′′} is injective. Equivalently

H_{r}(K)Q′′P_{H_{r}(\overline{Q}′′)K_{+}^{e}} = H_{r}(\overline{Q}′′)K_{+}^{e}.  (104)

Proof:

1. We show (101). By Corollary 4.2, we know that

H_{r}(K_{+}^{e})P_{H_{r}(Q_{+}^{e})} = H_{r}(Q_{+}^{e}).  (105)

Now H_{r}(K)P_{H_{r}(Q′)} = H_{r}(Q′) if and only if H_{r}(K)Q′′P_{H_{r}(Q′)Q′′} = H_{r}(Q′)Q′′. Since H_{r}(Q′′) is orthogonal to both H_{r}(K)Q′′ and H_{r}(Q′)Q′′, the last equality is equivalent to H_{r}(KQ′′)P_{H_{r}(Q′Q′′)} = H_{r}(Q′Q′′) or to

H_{r}(KQ′′)P_{H_{r}( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )} = H_{r}(\overline{Q}′′K_{+}^{e})P_{H_{r}( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )} = H_{r} ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) .

Thus, it suffices to show that

H_{r}(\overline{Q}′′K_{+}^{e})P_{H_{r}(Q_{+}^{e})} = H_{r}(Q_{+}^{e}),  (106)

and

H_{r}(\overline{Q}′′K_{+}^{e})P_{H_{r}(R^{e})} = H_{r}(R^{e}).  (107)

Using the orthogonality of H_{r}(K_{+}^{e}) and H_{r}(R^{e}), we compute

\begin{aligned}H_{r}(\overline{Q}′′K_{+}^{e})P_{H_{r}(R^{e})} & = & [H_{r}(\overline{Q}′′)K_{+}^{e} ⊕ H_{r}(K_{+}^{e})]P_{H_{r}(R^{e})} = H_{r}(\overline{Q}′′)K_{+}^{e}P_{H_{r}(R^{e})} \\ & = & H_{r}(\overline{Q}′′)P_{H_{r}(R^{e})} = H_{r}(R^{e}). \end{aligned}

Here we used (100). Similarly, using (105),

H_{r}(\overline{Q}′′K_{+}^{e})P_{H_{r}(Q^{e})} = [H_{r}(\overline{Q}′′)K_{+}^{e} ⊕ H_{r}(K_{+}^{e})]P_{H_{r}(Q^{e})} ⊃ H_{r}(K_{+}^{e})P_{H_{r}(Q^{e})} = H_{r}(Q^{e}).

As the opposite inclusion is trivially satisfied, we have the equality.

2. Since the subspaces H_{r}(K)Q′′ and H_{r}(Q′′) are orthogonal, we have H_{r}(K)Q′′|P_{H_{r}(Q′′)} = 0. Using this, as well as equality (101), we compute

\begin{aligned}H_{r}(K)Q′′P_{H_{r}(Q′Q′′)} & = & H_{r}(K)Q′′P_{H_{r}(Q′)Q′′⊕H_{r}(Q′′)} = H_{r}(K)Q′′[P_{H_{r}(Q′)Q′′} + P_{H_{r}(Q′′)}] \\ & = & H_{r}(K)Q′′P_{H_{r}(Q′)Q′′} = H_{r}(Q′)Q′′ \end{aligned}

This can be rewritten as

H_{r}(K)Q′′P_{H_{r}( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )} = H_{r}(Q′)Q′′.

To the last equality we apply the projection P_{H_{r}(R)}, noting that, for all f ∈ H_{+}^{2}, we have

fP_{H_{r}( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )}P_{H_{r}(R)} = fP_{H_{r}(R)}, to obtain

\begin{aligned}H_{r}(K)Q′′P_{H_{r}(R)} & = & H_{r}(K)Q′′P_{H_{r}( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )}P_{H_{r}(R)} \\ & = & H_{r}(Q′)Q′′P_{H_{r}(R)} = H_{r}(R). \end{aligned}

In the last line we used (98).

3. The inclusion H_{r}(K)Q′′ ⊂ H_{r} ( \begin{matrix}K_{−}Q_{+} & 0 \\ 0 & R \end{matrix} ) implies the equality

H_{r}(K)Q′′ = H_{r}(K)Q′′P_{H_{r}( \begin{matrix}K_{−}Q_{+} & 0 \\ 0 & R \end{matrix} )}.

So, using (102), we compute

H_{r}(K)Q′′π_{2} = H_{r}(K)Q′′P_{H_{r}( \begin{matrix}K_{−}Q_{+} & 0 \\ 0 & R \end{matrix} )}π_{2}
= H_{r}(K)Q′′[P_{H_{r}(K_{−}^{e}Q_{+}^{e})} + P_{H_{r}(R^{e})}]π_{2}
= H_{r}(K)Q′′P_{H_{r}(R^{e})}π_{2} = H_{r}(R^{e})π_{2} = H_{r}(R^{e}).

4. The proof is analogous to the proof of the first statement and we omit it.

The previous result can have many equivalent restatements. Intuitively, they all state that there is an excess of poles over zeros.

Corollary 5.6 Let W be a minimal, stable spectral factor. Then with the notation of Theorem 5.3,

1. The Toeplitz operator T_{K(Q′)^{∗}}^{r} = T_{(\overline{Q}′)^{∗}K_{−}^{e}}^{r} is injective.

2. We have

H_{+}^{2}Q′ + H_{−}^{2}K = L^{2}.  (108)

3. We have

H_{+}^{2}K ∩ H_{−}^{2}Q′ = {0}.  (109)

4. We have

H_{+}^{2}K ∩ H_{r}(Q′) = {0}.  (110)

Proof: By Lemma 5.3, we have the injectivity of the projection map H_{r}(Q′)Q′′|P_{H_{r}(K)Q′′} which is equivalent to the injectivity of the projection map H_{r}(Q′)|P_{H_{r}(K)}. We apply now Theorem 4.1 to infer (108), (109) and (110).

We proceed to prove an important geometric result whose control significance will be stated at the end of the proof.

Theorem 5.4 Let W be a minimal, stable spectral factor. Then, with the notation of Theorem 5.3, we have

1.

H_{r}(R)P_{H_{r}(K)Q′′} ⊂ H_{r}(Q′)Q′′P_{H_{r}(K)Q′′} ∩ H_{r}(\overline{Q}′′)K_{+}P_{H_{r}(K)Q′′}  (111)

2.

H_{r}(R^{e})P_{H_{r}(K)Q′′} = H_{r}(Q′_{2})Q′′P_{H_{r}(K)Q′′} = H_{r}(\overline{Q}′′_{2})\overline{Q}′′_{1}K_{+}^{e}P_{H_{r}(K)Q′′}  (112)

Proof:

1. We use the factorization Q′Q′′ = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) to conclude

H_{r}(Q′)Q′′ ⊕ H_{r}(Q′′) = H_{r}(Q′Q′′) = H_{r} ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) = H_{r}(Q_{+}^{e}) ⊕ H_{r}(R^{e}) ⊃ H_{r}(R^{e}).

Therefore

\begin{aligned}H_{r}(Q′)Q′′P_{H_{r}(K)Q′′} & = & H_{r}(Q′Q′′)P_{H_{r}(K)Q′′} \\ & = & H_{r} ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) P_{H_{r}(K)Q′′} ⊃ H_{r}(R^{e})P_{H_{r}(K)Q′′}. \end{aligned}

Similarly, we have KQ′′ = \overline{Q}′′K_{+}^{e} and hence

H_{r}(\overline{Q}′′)K_{+}^{e}P_{H_{r}(K)Q′′} = [H_{r}(\overline{Q}′)KQ′′ ⊕ H_{r}(\overline{Q}′′)K_{+}^{e}] P_{H_{r}(K)Q′′}
= [H_{r}(\overline{Q}′)\overline{Q}′′ ⊕ H_{r}(\overline{Q}′′)] K_{+}^{e}P_{H_{r}(K)Q′′} = H_{r}(\overline{Q}′\overline{Q}′′)K_{+}^{e}P_{H_{r}(K)Q′′}
= H_{r} ( \begin{matrix}Q_{−} & 0 \\ 0 & R \end{matrix} ) K_{+}^{e}P_{H_{r}(K)Q′′}
= [H_{r}(Q_{−}^{e})K_{+}^{e} ⊕ H_{r}(R^{e})] P_{H_{r}(K)Q′′} ⊃ H_{r}(R^{e})P_{H_{r}(K)Q′′}.

From these two inclusions we obtain the inclusion

H_{r}(R)P_{H_{r}(K)Q′′} ⊂ H_{r}(Q′)Q′′P_{H_{r}(K)Q′′} ∩ H_{r}(\overline{Q}′′)K_{+}P_{H_{r}(K)Q′′}.  (113)

2. We have Q′_{2}Q′′_{2} = ( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} ). Since Q′_{2}, Q are left coprime and Q′′_{2}, Q are right coprime,

we obtain the equivalence of the inner functions Q, R, Q′_{2}, Q′′_{2}. Now we consider the direct sum decomposition H_{r}(Q′_{2}Q′′) = H_{r}(Q′_{2})Q′′ ⊕ H_{r}(Q′′) which, by the orthogonality of H_{r}(K)Q′′ and H_{r}(Q′′), leads to H_{r}(Q′_{2})Q′′P_{H_{r}(K)Q′′} = H_{r}(Q′_{2}Q′′)P_{H_{r}(K)Q′′}. But we also have

\begin{aligned}H_{r}(Q′_{2}Q′′) & = & H_{r}(Q′_{2}Q′′_{2}Q′′_{1}) = H (( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} ) Q′′_{1}) \\ & = & H_{r}(Q^{e})Q′′_{1} ⊕ H_{r}(R^{e}) ⊃ H_{r}(R^{e}). \end{aligned}

Hence

H_{r}(Q′_{2})Q′′P_{H_{r}(K)Q′′} = H_{r}(Q′_{2}Q′′)P_{H_{r}(K)Q′′} ⊃ H_{r}(R^{e})P_{H_{r}(K)Q′′}.

Similarly, note that

H_{r}(\overline{Q}′_{2})\overline{Q}′′_{2}\overline{Q}′′_{1}K_{+}^{e} = H_{r}(\overline{Q}′_{2})\overline{Q}′′K_{+}^{e} = H_{r}(\overline{Q}′_{2})KQ′′.

So H_{r}(\overline{Q}′_{2})KQ′′ ⊂ Ker P_{H_{r}(K)Q′′} and hence

H_{r}(\overline{Q}′′_{2})\overline{Q}′′_{1}K_{+}^{e}P_{H_{r}(K)Q′′} = [H_{r}(\overline{Q}′_{2})\overline{Q}′′_{2}\overline{Q}′′_{1}K_{+}^{e} ⊕ H_{r}(\overline{Q}′′_{2})\overline{Q}′′_{1}K_{+}^{e}]P_{H_{r}(K)Q′′}
= [H_{r}(\overline{Q}′_{2})\overline{Q}′′_{2} ⊕ H_{r}(\overline{Q}′′_{2})]\overline{Q}′′_{1}K_{+}^{e}P_{H_{r}(K)Q′′}
= H_{r}(\overline{Q}′_{2}\overline{Q}′′_{2})\overline{Q}′′_{1}K_{+}^{e}P_{H_{r}(K)Q′′}
= [H_{r}(( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} ))\overline{Q}′′_{1}K_{+}^{e}] P_{H_{r}(K)Q′′}
= [H_{r}(Q^{e})(\overline{Q}′′_{1})^{e}K_{+}^{e} ⊕ H_{r}(R^{e})] P_{H_{r}(K)Q′′}
⊃ H_{r}(R^{e})P_{H_{r}(K)Q′′}

These two inclusions prove the inclusion

H_{r}(R^{e})P_{H_{r}(K)Q′′} ⊂ H_{r}(Q′_{2})Q′′P_{H_{r}(K)Q′′} ∩ H_{r}(\overline{Q}′′_{2})\overline{Q}′′_{1}K_{+}^{e}P_{H_{r}(K)Q′′}.  (114)

By Lemma 5.3, the projection P_{H_{r}(K)Q′′} is injective on H_{r}(R^{e}), which implies

dim(H_{r}(R^{e})P_{H_{r}(K)Q′′}) = dim(H_{r}(R^{e})).

Also, by Theorem 5.3, the inner functions R, Q′_{2}, \overline{Q}′′_{2} are equivalent and hence in particular

dim(H_{r}(R^{e})) = dim H_{r}(Q′_{2})Q′′ = dim H_{r}(\overline{Q}′′_{2})\overline{Q}′′_{1}K_{+}^{e}.

Therefore the inclusion (114) forces the equality (112).

For the case of external spectral factors, the inclusion (111) can be strengthened to give.

Corollary 5.7 Let W be a minimal, stable, external spectral factor. Then

H_{r}(R^{e})P_{H_{r}(K)Q′′} = H_{r}(Q′)Q′′P_{H_{r}(K)Q′′} = H_{r}(\overline{Q}′′)K_{+}^{e}P_{H_{r}(K)Q′′}.  (115)

Proof: By Theorem 5.4, we have

H_{r}(R^{e})P_{H_{r}(K)Q′′} = H_{r}(Q′)Q′′P_{H_{r}(K)Q′′} ∩ H_{r}(\overline{Q}′′)K_{+}^{e}P_{H_{r}(K)Q′′}.

But using the injectivity of the projection P_{H_{r}(K)Q′′} on the three subspaces and the fact that they all have the same dimension, the equality follows.

In a sense not made explicit in this paper, the space H_{r}(K)Q′′ can be used as a natural state space for a state space realization of the spectral factor W. This is based on the realization theory developed in Fuhrmann [1981] and we refer to it as the shift realization. Thus the subspaces H_{r}(Q′)Q′′P_{H_{r}(K)Q′′}, H_{r}(\overline{Q}′′)K_{+}P_{H_{r}(K)Q′′}, H_{r}(Q′_{2})Q′′P_{H_{r}(K)Q′′}, H_{r}(\overline{Q}′′_{2})\overline{Q}′′_{1}K_{+}^{e}P_{H_{r}(K)Q′′} and H_{r}(R^{e})P_{H_{r}(K)Q′′} are all special subspaces of H_{r}(K)Q′′, and it is natural to inquire as to their system theoretic significance. Without going into the details, we just state here that, with respect to the shift realization, the first two are the maximal output nulling, inner stabilizable subspace and the maximal output nulling, inner antistabilizable subspace respectively, whereas the last three are different representations of the maximal output nulling, reachability subspace. Moreover the inclusion in (111) can be shown to be actually an equality. A state space approach to these results will be given in Gombani and Fuhrmann [1998], while a functional approach to these characterizations will be presented in Fuhrmann [1998].

Sofar, we have bypassed the question of the existence of balanced factorizations. The following proposition shows that plenty of balanced factorizations exist.

Proposition 5.5 Let Q_{+} be an inner function and let R be an inner function equivalent to it. Then there exists a balanced factorization

( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) = Q′Q′′.  (116)

Proof: By the assumption of equivalence, there exist, necessarily inner, matrix functions T, S in H_{+}^{∞} such that the intertwining relation

Q_{+}T = SR  (117)

holds, with Q_{+}, S left coprime and T, R right coprime. Equation (117), see Fuhrmann [1981] for the details, gives rise to an H_{+}^{∞}-isomorphism Z : H_{r}(Q_{+}) ⟶ H_{r}(R), defined by

fZ = fT P_{H_{r}(R)}.

The isomorphism is in the sense that, for every ϕ ∈ H_{+}^{∞} and f ∈ H_{r}(Q_{+}), we have

fϕP_{+}T P_{H_{r}(R)} = fT P_{H_{r}(R)}ϕP_{+}.

The adjoint map, Z^{∗} : H_{r}(R) ⟶ H_{r}(Q_{+}), is given by

gZ^{∗} = gT^{∗}P_{+}

and we have, again for every ϕ ∈ H_{+}^{∞}, that

gϕ^{∗}P_{+}T^{∗}P_{+} = gP_{+}T^{∗}P_{+}ϕ^{∗}.

Next, we define a subspace of H_{r} ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) = H_{r}(Q_{+}^{e}) ⊕ H_{r}(R^{e}) by

V = {( gZ^{∗} g ) |g ∈ H_{r}(R)}.

Clearly, V is a coinvariant subspace. In fact, given ϕ ∈ H_{+}^{∞}, we compute

( gZ^{∗}P_{+} g ) ϕ^{∗}P_{+} = ( gT^{∗}P_{+}ϕ^{∗}P_{+} gϕ^{∗}P_{+} )
= ( gϕ^{∗}P_{+}T^{∗}P_{+} gϕ^{∗}P_{+} )
= ( g_{1}Z^{∗} g_{1} ) ∈ V,

where g_{1} = gϕ^{∗}P_{+} ∈ H_{r}(R). Thus, by Beurling’s theorem, there exists an inner function Q′′ for which V = H_{r}(Q′′). This implies the factorization (116).

We show that necessarily, the factorization (116) is balanced. Clearly, by construction, we have

\begin{aligned}Vπ_{1} & = & VP_{H_{r}(Q_{+}^{e})} = H_{r}(Q_{+}^{e}) \\ Vπ_{2} & = & VP_{H_{r}(R^{e})} = H_{r}(R^{e}). \end{aligned}  (118)

If Q′′, Q_{+}^{e} are not right coprime, we set Q′′ ∧_{R} Q_{+}^{e} = Q′′_{1}. Thus there exist a factorization Q′′ = Q′′_{2}Q′′_{1} and hence H_{r}(Q′′) = H_{r}(Q′′_{2})Q′′_{1} ⊕ H_{r}(Q′′_{1}). Clearly this implies that H_{r}(Q′′)π_{2} = H_{r}(Q′′_{2})Q′′_{1}π_{2} cannot be equal to H_{r}(R), in contradiction to (118). The other coprimeness conditions are proved similarly.

We can give now a characterization of minimal, stable, external spectral factors.

Theorem 5.5 There is a bijective correspondence between

1. Minimal, external p × m spectral factors.

2. Balanced factorizations of inner functions of the form

( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) ,  (119)

with R ≃ Q_{+} being an (m − m_{0}) × (m − m_{0}) inner function.

3. The set of all H_{+}^{∞}-isomorphisms from H_{r}(Q′′) onto H_{r}(R), with R ≃ Q_{+} being an (m − m_{0}) × (m − m_{0}) inner function.

Proof: Assume W is a minimal, stable, p × m external spectral factor. By Theorem 5.3, and using its notation, we have the existence of a factorization of the form (119) which, by Definition 5.3, is balanced. Conversely, let (119) be any balanced factorization. Such factorizations exist by Proposition 5.5. Define now W = W_{−}^{e}Q′. Clearly W is a stable spectral factor. Since W Q′′ = W_{+}^{e}, it has to be minimal. Clearly this correspondence is bijective.

From Proposition 5.5 it follows that to each such H_{+}^{∞}-isomorphism there corresponds an essentially unique balanced factorization. Conversely, the balanced factorization (119), can be written as Q′Q′′ = Q_{+}^{e}R. Using the coprimeness conditions, we can embed this equality in a doubly coprime factorization. This implies the existence of rational matrices L, M satisfying Q′′L = MR^{e}. Define a map Z : H_{r}(Q′′) ⟶ H_{r}(R^{e}) by fZ = fLP_{H_{r}(R^{e})}. Using the left coprimeness of Q′′, M and the right coprimeness of L, R^{e} which follow from the construction of a doubly coprime factorization, the map Z is indeed an H_{+}^{∞}-isomorphism. Note that the doubly coprime factorization is not unique, but the intertwining map Z is. Thus there exists a corresponding unique isomorphism between the set of balanced factorizations and the set of H_{+}^{∞}-isomorphisms from H_{r}(Q′′) onto H_{r}(R). Now, by Theorem 5.3 and using the fact that W is external, we have the equivalence of Q′′ and Q_{+}. Fix any H_{+}^{∞}-isomorphism of H_{r}(Q′′) and H_{r}(R^{e}) and the result follows by transitivity.

Theorem 5.5 leads immediately to a full parametrization of the set W^{m}.

Theorem 5.6 There is a bijective correspondence between

1. The set W^{m} of all p × m, minimal, stable spectral factors

2. Factorizations of the form

Q′_{1}Q′_{2}Q′′_{2}Q′′_{1} = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )

where Q_{+}^{e} = Q′_{1}QQ′′_{1} with all factors block diagonal of the form ( \begin{matrix}∗ & 0 \\ 0 & I_{m−m_{0}} \end{matrix} ), and

Q′_{2}Q′′_{2} = ( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} ) is a balanced factorization with R any (m − m_{0}) × (m − m_{0}) in-

ner function satisfying Q ≃ R. All inner functions are assumed to be normalized at infinity.

Proof: Follows from Theorems 5.3 and 5.5.

As a result of Theorem 5.5, it is clear that the existence of a p × m external spectral factor is dependent on m. Indeed we have the following.

Corollary 5.8 Let Φ be a rank m_{0} spectral function. Then there exists a p×m external spectral factor if and only if m−m_{0} is greater or equal to the number of nontrivial inner invariant factors of Q_{+}.

Proof: The inner function R has to be chosen to be equivalent to Q_{+}, thus necessarily it has the same number of nontrivial inner invariant factors as Q_{+} and its dimension has to be at least equal to that number.

We promised in section 3 to give a localized version of the phase function and this we proceed to do. To this end, we need first the following.

Definition 5.4 Let W_{1} and W_{2} be two internal spectral factors. We say that the W_{1} ≤ W_{2} if there exists an inner function Q such that W_{2} = W_{1}Q.

Lemma 5.4 Let W_{1}, W_{2} be two, minimal stable internal spectral factors. Let W_{i} = \overline{W}_{i}K_{i} be the DSS factorizations over H_{−}^{∞}. Assume W_{1} ≤ W_{2}, i.e. W_{1}Q = W_{2}, and let \overline{Q} be the inner function satisfying \overline{Q}K_{2} = K_{1}Q. Defining the local phase function by

T_{0}(W_{1}, W_{2}) = \overline{W} _{2}^{−L}W_{1},  (120)

we have

\overline{W}_{2}\overline{W} _{2}^{−L}W_{2} = W_{2}.  (121)
\overline{W}_{2}\overline{W} _{2}^{−L}W_{1} = W_{1}.  (122)
W_{2}W_{2}^{−L}W_{1} = W_{1}.  (123)
T_{0} = \overline{W} _{2}^{−L}W_{1} = K_{2}Q^{−1} = \overline{Q}^{−1}K_{1}.  (124)

Proof: By assumption, we have W_{1}Q = W_{2} and

\overline{W}_{2} = W_{2}K_{2}^{∗} = W_{1}QK_{2}^{∗} = W_{1}K_{1}^{∗}\overline{Q} = \overline{W}_{1}\overline{Q}

Since \overline{W} _{2}^{−L}\overline{W}_{2} = I, we have \overline{W}_{2}\overline{W} _{2}^{−L}\overline{W}_{2} = \overline{W}_{2}. Now

\overline{W}_{2}\overline{W} _{2}^{−L}W_{2} = \overline{W}_{2}\overline{W} _{2}^{−L}\overline{W}_{2}K_{2} = \overline{W}_{2}K_{2} = W_{2}.  (125)

This proves (121). (122) follows, using the fact that W_{1}Q = W_{2}. Substituting W_{2} = W_{1}Q in the equality W_{2}W_{2}^{−L}W_{2} = W_{2} and eliminating Q leads to (123). Finally, using (123), we compute

T_{0}(W_{1}, W_{2}) = \overline{W} _{2}^{−L}W_{1} = \overline{W} _{2}^{−L}(W_{2}W_{2}^{−L}W_{1}) = (\overline{W} _{2}^{−L}W_{2})(W_{2}^{−L}W_{1}) = K_{2}Q^{−1},

and use the equality K_{2}Q^{−1} = \overline{Q}^{−1}K_{1}.

Next we proceed to prove the following version of Proposition 4.4.

Proposition 5.6 Let Φ be a p × p, rank m_{0}, weakly coercive, spectral function, and let W_{1} ≤ W_{2} be two, minimal, stable, internal spectral factors. Let W_{i} = \overline{W}_{i}K_{i} be the respective DSS factorizations over H_{−}^{∞}. Let Q, \overline{Q} be the minimal McMillan degree inner functions satisfying \overline{Q}K_{2} = K_{1}Q. Then both Toeplitz operators T_{\overline{W} _{2}^{−L}W_{1}}^{c} = T_{K_{2}Q^{∗}}^{c} = T_{\overline{Q}^{∗}K_{1}}^{c} and T_{\overline{W} _{2}^{−L}W_{1}}^{r} are injective.

Proof: The proof follows that of Proposition 4.4. For the injectivity of T_{\overline{W} _{2}^{−L}W_{1}}^{c} we show an

explicit left inverse, namely the map f ↦ W_{1}^{−L}P_{+}\overline{W}_{2}f. Indeed, using (122), we compute

\begin{aligned}W_{1}^{−L}P_{+}\overline{W}_{2}P_{+}\overline{W} _{2}^{−L}W_{1}f & = & W_{1}^{−L}P_{+}\overline{W}_{2}\overline{W} _{2}^{−L}W_{1}f \\ & = & W_{1}^{−L}P_{+}W_{1}f = W_{1}^{−L}W_{1}f = f. \end{aligned}

The injectivity of T_{\overline{W} _{2}^{−L}W_{1}}^{r} follows from Proposition 4.2.

The previous result indicates that the definition of the phase function can be extended to all spectral factors. In fact, given a Lindquist-Picci pair W, \overline{W} of minimal spectral factors, then, with the notation of Theorem 5.3, we can define the associated phase function to be K(Q′)^{∗} = (\overline{Q}′)^{∗}K_{−}^{e}. Note that we already proved, in Corollary 5.6, the injectvity of the Toeplitz operator T_{K(Q′)^{∗}}^{r} = T_{(\overline{Q}′)^{∗}K_{−}^{e}}^{r}.

6 Minimal Markovian splitting subspaces

Each minimal, stable p × m spectral factor W has two, coprime, DSS factorizations. With each one there is associated a coinvariant subspace that is a natural state space for a shift based realization of W. If we choose to work with a left coprime factorization of W over H_{−}^{∞} then, as a consequence of Proposition 3.2, all minimal, stable spectral factors have the same left Douglas-Shapiro-Shields factor. This amounts to the fact that in a realization of any such spectral factor W, the matrices A, C can be chosen to be the same. This is the uniform choice of basis, see Caines and Delchamps [1980]. It turns out that for our purpose we have to work with right coprime DSS factorizations; in this case, the right denominators, and hence the associated state spaces, do not need to be the same, this fact is peculiar of the multivariable case. This enables us to study the zeros in purely geometric terms. We proceed to explain this association. By Proposition 5.2, there exists an essentially unique inner function Q′′ such that W_{+}^{e} = W Q′′. On the other hand, W has a right coprime DSS factorization W = \overline{W} K = \overline{W} K^{−∗}, with \overline{W} necessarily a minimal antistable spectral factor. Let H_{r}(K) = {H_{+}^{2}K}^{⊥} be the coinvariant subspace associated with the inner function K. We say that

X_{W} = H_{r}(K)Q′′  (126)

is the minimal Markovian splitting subspace associated with the minimal spectral factor W. Note that all the spaces H_{r}(K) carry isomorphic H_{+}^{∞}-module structures. This is due to the fact that the pole structure of all minimal, stable spectral factors is the same, and they differ only in their zero structure. The particular definition, given in (126) is a result of a normalization, where with W_{+} we associate the state space H_{r}(K_{+}^{e}). We denote by X^{m} the family of all minimal Markovian splitting subspaces associated to minimal, stable, spectral factors of size p × m. In an analogous way we can associate with the antistable factor \overline{W} the state space X\overline{W} = H_{r}(K^{∗})(\overline{Q}′)^{∗}. We denote by \overline{X}^{m} the set of all such spaces corresponding to factors of size p × m. At this point our definition of splitting subspaces is rather formal and devoid of any stochastic interpretation. But it turns out that this is the simplest way to transpose in the Hardy space setting the geometric structure of the stochastic domain, however we omit the details.

The following proposition organizes the information concerning the geometry of the splitting subspaces associated with an arbitrary minimal, stable spectral factor.

Proposition 6.1 . Let W be a minimal, stable spectral factor. Let W = \overline{W} K be its DSS factorization over H_{−}^{∞}. Let the inner functions Q′, Q′′, \overline{Q}′, \overline{Q}′′ be determined by Proposition 5.2. Then we have

1. The following are orthogonal direct sum decompositions of the coinvariant subspace H_{r}(\overline{Q}′KQ′′).

\begin{aligned}H_{r}(\overline{Q}′KQ′′) & = & H_{r}(Q_{−}^{e})K_{+}^{e} ⊕ H_{r}(R^{e}) ⊕ H_{r}(K_{+}^{e}) \\ & = & H_{r}(\overline{Q}′)\overline{Q}′′K_{+}^{e} ⊕ H_{r}(\overline{Q}′′)K_{+}^{e} ⊕ H_{r}(K_{+}^{e}) \\ & = & H_{r}(\overline{Q}′)KQ′′ ⊕ H_{r}(K)Q′′ ⊕ H_{r}(Q′′) \\ & = & H_{r}(K_{−}^{e})Q′Q′′ ⊕ H_{r}(Q′)Q′′ ⊕ H_{r}(Q′′) \\ & = & H_{r}(K_{−}^{e})Q_{+}^{e} ⊕ H_{r}(R^{e}) ⊕ H_{r}(Q_{+}^{e}) \end{aligned}  (127)

2. For L^{2} we have the following direct sum decomposition.

\begin{aligned}L^{2} & = & H_{+}^{2}Q′′ ⊕ H_{−}^{2}Q′′ \\ & = & H_{+}^{2}KQ′′ ⊕ H_{r}(K)Q′′ ⊕ H_{−}^{2}Q′′ \end{aligned}  (128)

3. Defining the L^{2} subspaces by

\begin{aligned}S_{W} & = & H_{+}^{2}Q′′ \\ \overline{S}_{W} & = & H_{−}^{2}KQ′′, \end{aligned}  (129)

we have

X_{W} = H_{r}(K)Q′′ = S_{W} ∩ \overline{S}_{W} .  (130)

The pair (S_{W} , \overline{S}_{W}) will be called an extended scattering pair for the spectral factor W.

4. We have

\begin{aligned}S_{W}^{⊥} & = & H_{−}^{2}Q′′ \\ \overline{S}_{W}^{⊥} & = & H_{+}^{2}KQ′′, \end{aligned}  (131)

and

L^{2} = \overline{S}_{W}^{⊥} ⊕ X_{W} ⊕ S_{W}^{⊥} .  (132)

5. The projection operators

R = H_{r}(K_{+}^{e})|P_{H_{r}(K)Q′′} : H_{r}(K_{+}^{e}) ⟶ H_{r}(K)Q′′  (133)

and

O = H_{r}(K)Q′′|P_{H_{r}(K_{−}^{e})Q′Q′′} : H_{r}(K)Q′′ ⟶ H_{r}(K_{−}^{e})Q′Q′′,  (134)

are bijective.

6. The projection operators

H_{r}(K)Q′′|P_{H_{r}(K_{+}^{e})} : H_{r}(K)Q′′ ⟶ H_{r}(K_{+}^{e})  (135)

and

H_{r}(K)Q′′|P_{H_{r}(K_{−}^{e})Q′Q′′} : H_{r}(K)Q′′ ⟶ H_{r}(K_{−}^{e})Q′Q′′  (136)

are bijective.

7. We have the following factorization

H_{r}(K_{+}^{e})|P_{H_{r}(K_{−}^{e})Q′Q′′} = (H_{r}(K_{+}^{e})|P_{H_{r}(K)Q′′}) · (H_{r}(K)Q′′|P_{H_{r}(K_{−}^{e})Q′Q′′}).  (137)

8. Let W_{i}, i = α, β be two minimal, stable spectral factors and let X_{W_{i}} = H_{r}(K_{i})Q′′_{i}. Then, with R_{i}, O_{i} defined as above, there exists a unique map Z : H_{r}(K_{β})Q′′_{β} ⟶ H_{r}(K_{α})Q′′_{α} which makes the following diagram commutative.

[figure: H_{r}(K_{−}^{e})Q_{+}^{e}; H_{r}(K_{β})Q′′_{β}; H_{r}(K_{α})Q′′_{α}; H_{r}(K_{+}^{e}); Z; O_{α}; R_{β}; O_{β}; R_{α}]

Diagram 7

thus

Z = R_{β}^{−1}R_{α} = O_{β}O_{α}^{−1}.  (138)

9. We have

∥fP_{H_{r}(K_{α})Q′′_{α}}∥^{2} ≤ ∥fP_{H_{r}(K_{β})Q′′_{β}}∥^{2} ∀f ∈ H_{r}(K_{+}^{e}).

if and only if

∥gP_{H_{r}(K_{α})Q′′_{α}}∥^{2} ≥ ∥gP_{H_{r}(K_{β})Q′′_{β}}∥^{2} ∀g ∈ H_{r}(K_{−}^{e})Q_{+}^{e}.

These inequalities characterize the contractivity of the map Z.

Proof:

1. From the commutativity of Diagram 3, we obtain the equality

\overline{Q}′KQ′′ = \overline{Q}′\overline{Q}′′K_{+}^{e} = K_{−}^{e}Q′Q′′ = Q_{−}^{e}K_{+}^{e}R^{e} = K_{−}^{e}Q_{+}^{e}R^{e},

and the result follows.

2. We use the equalities L^{2}Q′′ = L^{2}, H_{+}^{2} = H_{+}^{2}K ⊕ H_{r}(K) as well as the fact that multiplication by Q′′ is a unitary map in L^{2}.

3. We compute

\begin{aligned}S_{W} ∩ \overline{S}_{W} & = & H_{+}^{2}Q′′ ∩ H_{−}^{2}KQ′′ \\ & = & (H_{+}^{2} ∩ H_{−}^{2}K)Q′′ = H_{r}(K)Q′′ = X_{W} . \end{aligned}

4. Since S_{W} = H_{+}^{2}Q′′ and L^{2} = H_{+}^{2}Q′′ ⊕ H_{−}^{2}Q′′, we conclude that S_{W}^{⊥} = H_{−}^{2}Q′′. Similarly, \overline{S}_{W} = H_{−}^{2}KQ′′ implies \overline{S}_{W}^{⊥} = H_{+}^{2}KQ′′. Thus (132) is equivalent to the second direct sum decomposition in (128).

5. Since, using Theorem 5.3, the equivalence of K_{+}^{e} and K implies that the dimension of the state spaces H_{r}(K_{+}^{e}) and H_{r}(K)Q′′ is equal, it suffices to prove that the projection P_{H_{r}(K)Q′′}|H_{r}(K_{+}^{e}) is injective. Now

\begin{aligned}Ker P_{H_{r}(K)Q′′} & = & H_{+}^{2}KQ′′ ⊕ H_{r}(Q′′) \\ & = & H_{+}^{2}\overline{Q}′′K_{+}^{e} ⊕ H_{r}(Q′′). \end{aligned}

Since H_{r}(K_{+}^{e}) is orthogonal to H_{+}^{2}\overline{Q}′′K_{+}^{e}, we can compute

Ker H_{r}(K_{+}^{e})|P_{H_{r}(K)Q′′} = H_{r}(K_{+}^{e}) ∩ (H_{+}^{2}KQ′′ ⊕ H_{r}(Q′′))
= H_{r}(K_{+}^{e}) ∩ H_{r}(Q′′)
= H_{r}(K_{+}^{e} ∧_{R} Q′′) = {0}.

We clearly have K_{+}^{e} ∧_{R} Q′′, the greatest common right inner divisor of K_{+}^{e} and Q′′ equal to the identity because of the right coprimeness of these two inner functions. In this connection, see Theorem 5.2.

To prove the injectivity of (134), we show first that the following diagram commutes.

[figure: H_{r}(K_{−}^{e})Q′Q′′; \overline{H}_{r}((K_{−}^{e})^{∗}); H_{r}(K)Q′′; \overline{H}_{r}(K^{∗})\overline{Q}^{′∗}; P_{H_{r}(K)Q′′}; P_{\overline{H}_{r}(K^{∗})\overline{Q}^{′∗}}; Q^{′′∗}Q^{′∗}K_{−}^{∗}; Q^{′′∗}K^{∗}\overline{Q}^{′∗}]

Diagram 8

We note that, for f ∈ H_{+}^{2}, we have fP_{H_{r}(K)Q′′} = fQ^{′′∗}P_{+}K^{∗}P_{−}KQ′′. If f ∈ H_{r}(K_{−}^{e})Q′Q′′ then f = gQ′Q′′ with g ∈ H_{r}(K_{−}^{e}).

We compute

\begin{aligned}fP_{H_{r}(K)Q′′}(Q′′)^{∗}K^{∗}(\overline{Q}′)^{∗} & = & gQ′Q′′Q^{′′∗}P_{+}K^{∗}P_{−}KQ′′(Q′′)^{∗}K^{∗}(\overline{Q}′)^{∗} \\ & = & gQ′K^{∗}P_{−}(\overline{Q}′)^{∗}. \end{aligned}

Going the other way, and noting that P_{\overline{H}_{r}(K^{∗})(\overline{Q}′)^{∗}} restricted to H_{−}^{2} is given by h ↦ h\overline{Q}′P_{−}KP_{+}K^{∗}(\overline{Q}′)^{∗}, we compute

fQ^{′′∗}Q^{′∗}P_{H_{r}(K^{∗})(\overline{Q}′)^{∗}} = gQ′Q′′Q^{′′∗}Q^{′∗}K_{−}^{∗}\overline{Q}′P_{−}KP_{+}K^{∗}(\overline{Q}′)^{∗}
= gK_{−}^{∗}\overline{Q}′P_{−}KP_{+}K^{∗}(\overline{Q}′)^{∗}
= gQ′K^{∗}P_{−}KP_{+}K^{∗}(\overline{Q}′)^{∗}
= gQ′K^{∗}P_{−}(\overline{Q}′)^{∗},

for gQ′K^{∗}P_{−} ∈ \overline{H}_{r}(K^{∗}) and hence gQ′K^{∗}P_{−}KP_{+}K^{∗} = gQ′K^{∗}P_{−}P_{\overline{H}_{r}(K^{∗})} = gQ′K^{∗}P_{−}.

The bijectivity of P_{H_{r}(K)Q′′} : H_{r}(K_{+}^{e}) ⟶ H_{r}(K)Q′′ is equivalent to the bijectivity of P_{\overline{H}_{r}(K^{∗})(\overline{Q}′)^{∗}} : \overline{H}_{r}(K_{−}^{∗}) ⟶ \overline{H}_{r}(K^{∗})(\overline{Q}′)^{∗}, and this we proceed to prove. As in the first part,

\begin{aligned}Ker P_{\overline{H}_{r}(K^{∗})(\overline{Q}′)^{∗}}|\overline{H}_{r}(K_{−}^{∗}) & = & \overline{H}_{r}(K_{−}^{∗}) ∩ (\overline{H}_{r}((\overline{Q}′)^{∗}) ⊕ H_{−}^{2}K^{∗}(\overline{Q}′)^{∗}) \\ & = & \overline{H}_{r}(K_{−}^{∗}) ∩ \overline{H}_{r}((\overline{Q}′)^{∗}) = \{0\}. \end{aligned}

The last equality follows from the left coprimeness of K_{−}^{e} and \overline{Q}′.

6. Follows from the previous part by taking adjoints.

7. This follows from the three direct sum decompositions in (127). For F ∈ H_{r}(\overline{Q}′KQ′′), let

F = f_{1} + f_{2} + f_{3} = g_{1} + g_{2} + g_{3} = h_{1} + h_{2} + h_{3}

be the representations with respect to these direct sum decompositions. Clearly, we have f_{1} = g_{1} and g_{3} = h_{3}. Assume now that F ∈ H_{r}(K_{+}^{e}), so we have F = f_{3} = g_{2}+g_{3} = g_{2}+h_{3}. This implies g_{2} = f_{3}P_{H_{r}(K)Q′′}. Now g_{2} = h_{1} + h_{2} and so h_{1} = g_{2}P_{H_{r}(K_{−}^{e})Q′Q′′}. But f_{3} = g_{2} + h_{3} = h_{1} + h_{2} + h_{3} and so we have also h_{1} = f_{3}P_{H_{r}(K_{−}^{e})Q′Q′′}.

8. By part 5, the maps R_{1}, R_{2} are invertible. Computing the adjoints of the maps O_{1}, O_{2} we obtain O_{i}^{∗} = P_{H_{r}(K_{i})Q′′_{i}}|H_{r}(K_{−}^{e})Q′_{i}Q′′_{i} and these are also invertible. Hence so are the O_{i}. By part 7 we have R_{1}O_{1} = R_{2}O_{2}, for both are factorizations of P_{H_{r}(K_{−}^{e})Q′Q′′} : H_{r}(K_{+}^{e}) ⟶ H_{r}(K_{−}^{e})Q′Q′′. This implies the equality Z = R_{2}^{−1}R_{1} = O_{2}O_{1}^{−1}. This part of the proof is adapted from Lindquist and Picci [1991].

9. Assume ∥fP_{H_{r}(K_{1})Q′′_{1}}∥^{2} ≤ ∥fP_{H_{r}(K_{2})Q′′_{2}}∥^{2} for all f ∈ H_{r}(K_{+}^{e}), i.e. ∥fR_{1}∥ ≤ ∥fR_{2}∥. Setting g = fR_{2}, we get ∥gZ∥ = ∥gR_{2}^{−1}R_{1}∥ ≤ ∥g∥, i.e. Z is contractive. But then so is Z^{∗}, which implies ∥fZ^{∗}∥ = ∥fO_{1}^{−∗}O_{2}^{∗}∥ ≤ ∥f∥, for all f ∈ H_{r}(K_{−}^{e})Q_{+}^{e}. Setting g = fO_{1}^{∗}, we get ∥fO_{2}^{∗}∥ ≤ ∥fO_{1}^{∗}∥, i.e. ∥fP_{H_{r}(K_{2})Q′′_{2}}∥^{2} ≤ ∥fP_{H_{r}(K_{1})Q′′_{1}}∥^{2} for all f ∈ H_{r}(K_{−}^{e})Q_{+}^{e}.

In (129) we have defined the extended scattering pairs; this is because in the literature, see Lindquist and Picci [1991], a different object is called scattering pair. We briefly introduce some basic facts about it.

First, we need a simple lemma. Let x, y ∈ L_{r}^{2}: we say that x and y are pointwise orthogonal if x(iω)y^{∗}(iω) = 0 for all ω ∈ R I . The pointwise orthogonal complement Y of a subspace X is the set of all vectors y ∈ L_{r}^{2} which are pointwise orthogonal to all vectors x ∈ X. Let K be an inner function. We set π_{K}^{⊥} to be the orthogonal projection of L^{2} onto the pointwise orthogonal complement of H_{r}(K) and we set π_{K} := I − π_{K}^{⊥} We also set L_{K}^{2} := \overline{span}\{e^{iωs}H_{r}(K); s ∈ R I \}.

Lemma 6.1 We have the following relations:

1.

Im π_{K} = L_{K}^{2}  (139)

and

2.

Kπ_{K} = π_{K}K  (140)

Proof:

1. We show that

Im π_{K}^{⊥} = \overline{span}\{e^{iωs}H_{r}(K); s ≥ 0\}^{⊥}

If f ∈ Im π_{K}^{⊥}, then f is pointwise orthogonal to any g ∈ H_{r}(K); that is g(iω)f(iω)^{∗} ≡ 0. But then also e^{iωs}g(iω)f(iω)^{∗} ≡ 0 for all s ∈ R I , which implies that f is orthogonal to L_{K}^{2}. Conversely, if f ∈ (L_{K}^{2})^{⊥}, it means that for each g ∈ L_{K}^{2}, it is ∫ II g(iω)f(iω)^{∗}d(ω) = 0. In particular, since L_{K}^{2} is invariant, we also have ∫ II e^{iωs}g(iω)f(iω)^{∗}d(ω) = 0 which implies that g(iω)f(iω)^{∗} ≡ 0

2. Let V_{r} = {v_{1}^{r}, ..., v_{k}^{r}} be a row basis for H_{r}(K). It is well known that there exist column vectors c_{i}^{r} in CI ^{m} such that K = I + ∑_{i=1}^{r} c_{i}^{r}v_{i}^{r}. Similarly if V_{c} = {v_{1}^{c}, ..., v_{k}^{c}} is a basis for H_{c}(K) then K = I + ∑_{i=1}^{r} v_{i}^{c}c_{i}^{c} for suitable row vectors c_{i}^{c}. We now exploit the fact that for f ∈ H_{r}(K), we have fπ_{K} = f and similarly, for g ∈ H_{c}(K), we have π_{K}g = g. Then

K(s)π_{K} = π_{K} + ∑_{i=1}^{r} c_{i}^{r}v_{i}^{r}π_{K} = π_{K} + ∑_{i=1}^{r} c_{i}^{r}v_{i}^{r}
= π_{K} + ∑_{i=1}^{r} v_{i}^{c}c_{i}^{c} = π_{K} + π_{K} ∑_{i=1}^{r} v_{i}^{c}c_{i}^{c} = π_{K}K(s)

Proposition 6.2 Let W be a minimal, stable spectral factor. Let W = \overline{W} K be its DSS factorization over H_{−}^{∞}. Let the inner functions Q′, Q′′, \overline{Q}′, \overline{Q}′′ be determined by Proposition 5.2. Then we have

1. Defining the L_{K}^{2} subspaces by

\begin{aligned}Ŝ_{W} & = & H_{+}^{2}π_{K}Q′′ \\ ˆ\overline{S}_{W} & = & H_{−}^{2}π_{K}KQ′′, \end{aligned}  (141)

we have

X_{W} = H_{r}(K)Q′′ = Ŝ_{W} ∩ ˆ \overline{S}_{W} .  (142)

The pair (Ŝ_{W} , ˆ \overline{S}_{W}) will be called a scattering pair for the spectral factor W.

2. We denote by Ŝ_{W}^{⊥} the orthogonal complement of Ŝ_{W} in L_{K}^{2}. Similarly, we define ˆ \overline{S}_{W}^{⊥} to be the orthogonal complement in L_{K}^{2} of ˆ \overline{S}_{W}. Then we have

\begin{aligned}Ŝ_{W}^{⊥} & = & H_{−}^{2}π_{K}Q′′ \\ ˆ\overline{S}_{W}^{⊥} & = & H_{+}^{2}Kπ_{K}Q′′, \end{aligned}  (143)

and

L_{K}^{2} = ˆ \overline{S}_{W}^{⊥} ⊕ X_{W} ⊕ Ŝ_{W}^{⊥} .  (144)

Proof:

1. We compute, in view of Lemma 6.1

\begin{aligned}Ŝ_{W} ∩ ˆ \overline{S}_{W} & = & H_{+}^{2}π_{k}Q′′ ∩ H_{−}^{2}π_{k}KQ′′ = (H_{+}^{2}π_{k} ∩ H_{−}^{2}Kπ_{k})Q′′ \\ & = & [(H_{r}(K) ⊕ H^{2}Kπ_{k}) ∩ (H_{−}^{2}π_{k} ⊕ H_{r}(K)]Q′′ = H_{r}(K)Q′′ = X_{W} . \end{aligned}

2. Since Ŝ_{W} = H_{+}^{2}π_{k}Q′′ and L_{K}^{2} = H_{+}^{2}π_{k}Q′′ ⊕ H_{−}^{2}π_{k}Q′′, again in view of Lemma 6.1 we

conclude that Ŝ_{W}^{⊥} = H_{−}^{2}π_{k}Q′′. Similarly, ˆ \overline{S}_{W} = H_{−}^{2}π_{k}KQ′′ implies ˆ \overline{S}_{W}^{⊥} = H_{+}^{2}π_{k}KQ′′. Thus

L_{K}^{2}Q′′ = Ŝ_{W} ⊕ Ŝ_{W}^{⊥} = H_{+}^{2}π_{k}Q′′ ⊕ Ŝ_{W}^{⊥} = H_{+}^{2}π_{k}KQ′′ ⊕ H_{r}(K)π_{k}Q′′ ⊕ Ŝ_{W}^{⊥}
= H_{+}^{2}π_{k}KQ′′ ⊕ H_{r}(K)Q′′ ⊕ Ŝ_{W}^{⊥} = ˆ \overline{S}_{W}^{⊥} ⊕ H_{r}(K)Q′′ ⊕ Ŝ_{W}^{⊥}

The above proposition illustrates quite clearly the difference of our approach with respect to the one of Lindquist and Picci. We simply take ”larger” spaces to represent our scattering pair: since the main use of scattering pairs is in the state space they generate by their intersection, the results we are obtaining are exactly the same. But, as can be seen comparing the equivalent results in Propositions 6.1 and 6.2 the proofs are simpler. So, in the following, a scattering pair will implicitly assumed to be extended. For the scattering theory origins of scattering pairs, see Lax and Phillips [1967].

We proceed with a study of the zero structure of the set W^{m} of minimal, stable spectral factors. This is done mainly through a geometric study of the associated set X^{m} of minimal Markovian splitting subspaces. Since in our approach the splitting subspaces have a unified representation, given by (126), the study reduces to a great extent to the arithmetic of inner functions.

We are interested mainly in the parametrization of the set W^{m}, for m_{0} ≤ m ≤ n + p. Moreover, the set W^{m} or the corresponding set X^{m} carry with them a natural partial order. In the internal case, i.e. m = m_{0}, this partial order is closely related to the factorization of certain inner functions and hence also, see Fuhrmann [1995], to the set of solutions of a certain homogeneous algebraic Riccati equation.

We feel it is instructive to begin our study of the set W^{m} in the special case where the spectral function is of full rank and coercive and the factors are internal, i.e. m = m_{0} = p. We denote this set by W^{p}. This case shows the general scheme, and also clarifies the new concepts that have to be introduced for the analysis of the general case.

Without loss of generality, we can assume the normalization Φ(∞) = I. Thus all regular spectral factors can be assumed to satisfy W (∞) = I as well. This set has been fully studied, e.g. Fuhrmann [1995]. The inner functions Q_{−}, Q_{+}, K_{−}, K_{+} are defined as before. Heuristically, we shall say, given spectral factors W_{1} and W_{2}, that W_{1} ≤ W_{2} if W_{2} is closer to being maximum phase. This can be expressed precisely by saying W_{2} = W_{1}Q for some inner function Q, see the Definition 5.4. So, if W_{1}Q′′_{i} = W_{+}, this can be expressed also by saying that Q′′_{2} is a right factor of Q′′_{1}. Equivalently, Q′_{1} is a left factor of Q′_{2}. This leads to other, geometric, characterizations that turn out to be useful in the generalization to the non regular case. Moreover, given an inner function Q_{+}, the set of left inner factors of Q_{+} is clearly partially ordered with Q_{1} ≤ Q_{2} if Q_{1} is a left factor of Q_{2}. In this set Q_{+} is clearly the maximal element and I is the minimal. This order has been expressed in Fuhrmann [1995] via the set of nonnegative definite solutions of a homogeneous Riccati equation. It is not surprising that all these orders are equivalent. We express this equivalence in the following.

The set W^{p} can be parametrized in several different ways. In fact, by the results of section 3, given a normalized, minimal, stable spectral factor, there exists a normalized factorization (i.e. both factors normalized and inner) Q_{+} = Q′Q′′ such that

\begin{aligned}W_{−}Q′ & = & W \\ W Q′′ & = & W_{+} \end{aligned}  (145)

So normalized factorizations of Q_{+} are in a bijective correspondence with the elements of the set W^{p}, i.e. they parametrize X^{p}. Clearly, if Q′_{i}Q′′_{i} are the factorizations of Q_{+} associated with W_{i}, then W_{1} ≤ W_{2} if and only if Q′_{1} is a left factor of Q′_{2}. On the other hand, see Fuhrmann

[1995] for the details, if Q_{+} = ( \begin{matrix}A & B \\ −B^{∗}X_{+} & I \end{matrix} ) with the realization minimal and X_{+} the unique

positive definite solution of the homogeneous Riccati equation

A^{∗}X + XA + XBB^{∗}X = 0,  (146)

then an arbitrary normalized right inner factor of Q_{+} is given by Q = ( \begin{matrix}A & B \\ −B^{∗}X & I \end{matrix} ), where

X is a nonnegative definite solution of the Riccati equation (146). We denote by P the set of nonnegative definite solutions of this Riccati equation. P has a natural partial order induced by the standard order on Hermitian matrices.

We want to remark that, for a minimal spectral factor W that satisfies (145), the factorization Q_{+} = Q′Q′′ represents a parametrization of the set of stable and antistable zeros of W. In fact Q′′ determines the set of stable zeros of W that can still be moved to the right half plane, via multiplication by Q′′, whereas Q′ determines the set of antistable zeros of W, that is zeros of W_{−} that have been moved already to the right half plane, via multiplication by Q′. This issue will be addressed formally in a subsequent paper. For a full parametrization of the set of all minimal spectral factors, see Fuhrmann [1995].

There is a third order that can be considered and it is a geometric one. It is a variation on an order introduced by Lindquist and Picci [1991]. For each W ∈ W^{p} there is a unique right coprime DSS factorization W = \overline{W} K with K normalized inner and \overline{W} ∈ H_{−}^{∞}. If W Q′′ = W_{+} we associate with the spectral factor W the state space X_{W} = H_{r}(K)Q′′. We denote by X^{p} the set of these spaces. We say X_{W_{1}} ≤ X_{W_{2}} if

∥fP_{X_{W_{1}}}∥ ≤ ∥fP_{X_{W_{2}}}∥, ∀f ∈ X_{W_{+}}.  (147)

Of course we have X_{W_{+}} = H_{r}(K_{+}).

In the next theorem we show that all these partial orders are equivalent.

Theorem 6.1 Let Φ be a full rank, normalized, coercive spectral density. Let W^{p} be the set of all minimal, stable internal spectral factors of Φ, let X^{p} be the set of all state spaces corresponding to these factors. Let the inner function Q_{+} be defined as before and assume it has a

minimal realization Q_{+} = ( \begin{matrix}A & B \\ −B^{∗}X_{+} & I \end{matrix} ) with X_{+} the positive definite solution of the Riccati

equation

A^{∗}X + XA + XBB^{∗}X = 0.

Let P be the set of all nonnegative definite solutions of this Riccati equation. Let W_{α}, W_{β} ∈ W^{p}. Then the following conditions are equivalent.

1.

W_{1}(s)W_{1}(s)^{∗} ≤ W_{2}(s)W_{2}(s)^{∗},  Re s > 0.  (148)

2. For some inner function Q, we have W_{2} = W_{1}Q.

3. For some inner function Q, we have QQ′′_{2} = Q′′_{1}.

4.

Q′′_{1}(s)^{∗}Q′′_{1}(s) ≤ Q′′_{2}(s)^{∗}Q′′_{2}(s),  Re s > 0.  (149)

5. With H_{r}(K_{1})Q′′_{1}, H_{r}(K_{2})Q′′_{2} the state spaces associated with W_{1}, W_{2} respectively, we have

∥fP_{H_{r}(K_{1})Q′′_{1}}∥ ≤ ∥fP_{H_{r}(K_{2})Q′′_{2}}∥,  ∀f ∈ H_{r}(K_{+})  (150)

6. We have

∥gP_{H_{r}(Q′′_{1})}∥ ≥ ∥gP_{H_{r}(Q′′_{2})}∥,  ∀g ∈ H_{r}(Q_{+})  (151)

7. Q′′_{1}, Q′′_{2} have unique representations of the form Q_{i} = ( \begin{matrix}A & B \\ −B^{∗}X_{i} & I \end{matrix} ) with X_{i} nonnegative

definite solutions of the Riccati equation satisfying X_{1} ≥ X_{2}.

Proof: We prove the following implications:

(1) ⇔ (2) Assume statement (1) holds. This implies ∥ξW_{1}(s)∥ ≥ ∥ξW_{2}(s)∥, in Re s > 0, and hence Q(s) = W_{1}(s)^{−1}W_{2}(s) has at most a finite number of removable singularities in the right half plane. Thus Q(s) is analytic and contractive in the right half plane. Moreover, as the W_{i} are spectral factors, we have W_{1}(s)W_{1}(s)^{∗} = W_{2}(s)W_{2}(s)^{∗} on the boundary, i.e. Q(s) is unitary on the imaginary axis, which means it is inner. So (2) holds. Conversely, W_{2} = W_{1}Q implies that, for every s in the open right half plane,

W_{2}(s)W_{2}(s)^{∗} = W_{1}(s)Q(s)Q(s)^{∗}W_{1}(s)^{∗} ≤ W_{1}(s)W_{1}(s)^{∗}

as Q(s), being inner, is contractive in the open right half plane.

(3) ⇒ (5) Assume QQ′′_{2} = Q′′_{1} holds. We clearly have H_{r}(Q′′_{1}) = H_{r}(QQ′′_{2}) = H_{r}(Q)Q′′_{2} ⊕ H_{r}(Q′′_{2}) and this implies that for every f ∈ H(K_{+})

∥fP_{H_{r}(Q′′_{1})}∥ ≥ ∥fP_{H_{r}(Q′′_{2})}∥.  (152)

Note that we have K_{i}Q′′_{i} = \overline{Q}′′_{i}K_{+} and hence

H_{r}(K_{i})Q′′_{i} ⊂ H_{r}(K_{i})Q′′_{i} ⊕ H_{r}(Q′′_{i}) = H_{r}(K_{i}Q′′_{i}) = H_{r}(\overline{Q}′′_{i}K_{+}) = H_{r}(\overline{Q}′′_{i})K_{+} ⊕ H_{r}(K_{+}).

This implies the equality

∥f∥^{2} = ∥fP_{H_{r}(K_{i})Q′′_{i}}∥^{2} + ∥fP_{H_{r}(Q′′_{i})}∥^{2},

for all f ∈ H_{r}(K_{+}). As a consequence of (152), we conclude that statement (2) implies ∥fP_{H_{r}(K_{1})Q′′_{1}}∥ ≤ ∥fP_{H_{r}(K_{2})Q′′_{2}}∥.

(2) ⇔ (3) We have W_{1}Q′′_{1} = W_{+} = W_{2}Q′′_{2}. Since W_{2} = W_{1}Q_{1}, it follows that W_{1}Q′′_{1} = W_{1}QQ′′_{2}, and, by the invertibility of W_{1}, that Q′′_{1} = QQ′′_{2}. Conversely, assume QQ′′_{2} = Q′′_{1}. We compute

W_{2}Q′′_{2} = W_{+} = W_{1}Q′′_{1} = W_{1}QQ′′_{2},

which clearly implies the equality W_{2} = W_{1}Q.

(3) ⇔ (4) Inequality (149) follows from QQ′′_{2} = Q′′_{1} by the contractivity of Q in the open right half plane.

Conversely, inequality (149) shows that, with the exception of a finite number of points, the function Q = Q′′_{1}(Q′′_{2})^{−1} is analytic and contractive in the open right half plane. Thus the singularities are removable. As Q has unitary boundary values, it is necessarily inner.

(2) ⇔ (6) The factorizations Q_{+} = Q′_{1}Q′′_{1} = Q′_{2}Q′′_{2} = Q′_{1}QQ′′_{2} show that Q′′_{1} = QQ′′_{2}. We have therefore the inclusions H_{r}(Q′′_{2}) ⊂ H_{r}(Q′′_{1}) ⊂ H_{r}(Q_{+}). This implies

∥gP_{H_{r}(Q′′_{1})}∥ ≥ ∥gP_{H_{r}(Q′′_{2})}∥,  ∀g ∈ H_{r}(Q_{+}).  (153)

To prove the converse, note that H_{r}(Q′′_{i}) ⊂ {H_{+}^{2}Q_{+}}^{⊥}. Hence it follows that the inequality ∥gP_{H_{r}(Q′′_{1})}∥ ≥ ∥gP_{H_{r}(Q′′_{2})}∥ holds for all g ∈ H_{+}^{2}. We conclude that H_{r}(Q′′_{2}) ⊂ H_{r}(Q′′_{1}) and therefore Q′′_{1} = QQ′′_{2} for some inner function Q.

(3) ⇔ (7) This has been proved in Fuhrmann [1995].

(5) ⇔ (6) Assume ∥fP_{H_{r}(K_{1})Q′′_{1}}∥ ≤ ∥fP_{H_{r}(K_{2})Q′′_{2}}∥ for all f ∈ H_{r}(K_{+}). By a previous computation, this is equivalent to ∥fP_{H_{r}(Q′′_{1})}∥ ≥ ∥fP_{H_{r}(Q′′_{2})}∥ for f ∈ H_{r}(K_{+}). Now, if M ⊂ N are subspaces of a Hilbert space and P_{M}, P_{N} the respective orthogonal projections, then we have P_{M} = P_{M}P_{N}. We apply this in the following way. We clearly have Q′_{i}Q′′_{i} = Q_{+} and therefore H_{r}(Q′′_{i}) ⊂ H_{r}(Q_{+}). So P_{H_{r}(Q′′_{i})} = P_{H_{r}(Q′′_{i})}P_{H_{r}(Q_{+})}. Now, we claim that

H_{r}(K_{+})P_{H_{r}(Q_{+})} = H_{r}(Q_{+}).  (154)

To see this we consider the inverse of the phase function T_{0}^{−1} = W_{−}^{−1}\overline{W}_{+} = Q_{+}K_{+}^{∗}. The first factorization is a left Wiener-Hopf factorization with trivial factorization indices. Hence the Toeplitz operator T_{W_{−}^{−1}\overline{W}_{+}}^{r} = T_{Q_{+}K_{+}^{∗}}^{r} is invertible. Applying Theorem 4.1, we conclude that (154) holds. Now (152) implies

∥fP_{H_{r}(Q_{+})}P_{H_{r}(Q′′_{1})}∥ ≥ ∥fP_{H_{r}(Q_{+})}P_{H_{r}(Q′′_{2})}∥.

and hence that

∥gP_{H_{r}(Q′′_{1})}∥ ≥ ∥gP_{H_{r}(Q′′_{2})}∥,

for all g ∈ H_{r}(Q_{+}).

(6) ⇔ (3) From (6), using the factorizations Q_{+} = Q′_{i}Q′′_{i}, we conclude that

H_{r}(Q′_{1})Q′′_{1} = H_{r}(Q_{+}) ∩ H_{+}^{2}Q′′_{1} = Ker H_{r}(Q_{+})|P_{H_{r}(Q′′_{1})}
⊂ Ker P_{H_{r}(Q′′_{1})}|H_{r}(Q_{+}) = H_{r}(Q_{+}) ∩ H_{+}^{2}Q′′_{1}
= H_{r}(Q′_{2})Q′′_{2}.

This implies that Q′′_{1} = QQ′′_{2} for some inner function Q, and hence also Q′_{2} = Q′_{1}Q. So W_{2} = W_{−}Q′_{2} = W_{−}Q′_{1}Q = W_{1}Q, i.e. W_{1} ≤ W_{2}.

We pass now to the study of the set W^{m} of minimal, stable, p × m spectral factors in the general case (m_{0} ≤ m ≤ m_{0} + p), which is the central theme of this paper. We denote by W^{m_{0}} the set of all minimal, stable p × m_{0} internal spectral factors of Φ. We also denote by \overline{W}^{m} the set of all minimal, antistable p × m spectral factors of Φ. Again W^{m} can be partially ordered. Unhappily, defining W_{1} ≤ W_{2} if W_{2} = W_{1}Q for some inner function Q is too coarse an order, and it can be refined. Thus we proceed differently, defining a partial order in the sets W^{m}, \overline{W}^{m}, X^{m} and \overline{X}^{m}.

Definition 6.1 Let W_{1}, W_{2} be minimal, stable, p×m spectral factors and let X_{i} = H_{r}(K_{i})Q′′_{i} ∈ X^{m}, i = 1, 2 be the respective minimal Markovian splitting subspaces. Let W_{i} = \overline{W}_{i}K_{i} be the right coprime DSS factorizations over H_{−}^{∞}. Let \overline{Q}_{i} be defined via Diagram 3.

1. We say that W_{α} ≤ W_{β} if

W_{α}(s)W_{α}(s)^{∗} ≥ W_{β}(s)W_{β}(s)^{∗},  Re s > 0.  (155)

2. If \overline{W}_{α}, \overline{W}_{β} ∈ \overline{W}^{m}, we say that \overline{W}_{α} ≤ \overline{W}_{β} if

\overline{W}_{α}(s)\overline{W}_{α}(s)^{∗} ≥ \overline{W}_{β}(s)\overline{W}_{β}(s)^{∗},  Re s > 0.  (156)

3. We say that X_{1} ≤ X_{2} if

∥fP_{X_{1}}∥ ≤ ∥fP_{X_{2}}∥, ∀f ∈ H_{r}(K_{+}^{e}).  (157)

Here P_{X} denotes the orthogonal projection of H_{+}^{2} onto X.

4. Given two minimal markovian splitting subspaces \overline{X}_{i} = \overline{H}_{r}(K_{i}^{∗})(\overline{Q}′_{i})^{∗} ∈ \overline{X}^{m}, we say that

\overline{X}_{1} ≤ \overline{X}_{2} if

∥hP_{\overline{X}_{1}}∥ ≥ ∥hP_{\overline{X}_{2}}∥, ∀h ∈ \overline{H}_{r}((K_{−}^{e})^{∗}).  (158)

Here P\overline{X} denotes the orthogonal projection of H_{−}^{2} onto \overline{X}.

The definion of the partial order in the set W is customary, see Anderson [1971]. We point out the inversion of the direction of the inequality, compared with (157). Also, we note that, since multiplication by inner functions is a unitary map in L^{2} and therefore preserves orthogonality, (158) is equivalent to

∥fP_{H_{r}(K_{1})Q′′_{1}}∥ ≥ ∥fP_{H_{r}(K_{2})Q′′_{2}}∥, ∀f ∈ H_{r}(K_{−}^{e})Q′Q′′.  (159)

We recall that, assuming W_{−} is p × m_{0} and of full column rank, the inner functions Q_{−}, Q_{+}, K_{−}, K_{+} are all m_{0} × m_{0}, whereas the spectral factors we study are all of size p × m. We also recall that we have defined π_{1}, see definition 5.1, to be the projection on the first m_{0} components. For following the statement and proof of the next theorem, it is advisable to refer to Diagram (3).

Theorem 6.2 Let Φ be a rank m_{0}, spectral function, having no zeros on the extended imaginary axis. Let W^{m} be the set of all minimal, stable spectral factors of Φ, let X^{m} be the set of all minimal Markovian splitting subspaces corresponding to these factors. Let W_{α}, W_{β} ∈ W^{m}. Then the following statements are equivalent.

1.

W_{α} ≤ W_{β}.  (160)

2.

\overline{W}_{α} ≤ \overline{W}_{β}.  (161)

3.

π_{1}Q′′_{α}(s)^{−1}Q′′_{α}(s)^{−∗}π_{1} ≥ π_{1}Q′′_{β}(s)^{−1}Q′′_{β}(s)^{−∗}π_{1},  Re s > 0.  (162)

4.

π_{1}Q′′_{α}(s)^{∗}Q′′_{α}(s)π_{1} ≤ π_{1}Q′′_{β}(s)^{∗}Q′′_{β}(s)π_{1},  Re s > 0.  (163)

5. We have

∥P_{H_{c}((Q′′_{α})^{∗})}g∥ ≥ ∥P_{H_{c}((Q′′_{β})^{∗})}g∥,  ∀g ∈ \overline{H}_{c}((Q_{+}^{e})^{∗})  (164)

6. We have

∥gP_{H_{r}(Q′′_{α})}∥ ≥ ∥gP_{H_{r}(Q′′_{β})}∥,  ∀g ∈ H_{r}(Q_{+}^{e})  (165)

7. With the maps T_{i} defined by

T_{i} = H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{i})}  i = α, β  (166)

there exists a unique, contractive map Y for which T_{α}Y = T_{β}.

8. We have

∥fP_{H_{r}(K_{α})Q′′_{α}}∥ ≤ ∥fP_{H_{r}(K_{β})Q′′_{β}}∥,  ∀f ∈ H_{r}(K_{+}^{e})  (167)

9. Let the maps R_{i} : H_{r}(K_{+}^{e}) ⟶ H_{r}(K_{i})Q′′_{i} be defined by (133) for i = α, β. The map

Z : H_{r}(K_{β})Q′′_{β} ⟶ H_{r}(K_{α})Q′′_{α}

defined by Z = R_{β}^{−1}R_{α} is contractive.

Proof:

(1) ⇔ (2) Since W_{i} = \overline{W}_{i}K_{i}, we have

W_{i}W_{i}^{∗} = \overline{W}_{i}K_{i}K_{i}^{∗}\overline{W} _{i}^{∗} = \overline{W}_{i}\overline{W} _{i}^{∗}

which clearly implies the equivalence.

(1) ⇔ (3) We note that, on the imaginary axis, we have W_{α} = W_{+}^{e}(Q′′_{α})^{∗}, which clearly has a meromorphic extension to the right half plane, given by W_{α}(s) = W_{+}^{e}(s)Q′′_{α}(s)^{−1}. Assume W_{α} ≤ W_{β}, i.e. inequality (155) holds. Let W_{+}^{−L}(s) be a left inverse of W_{+}(s). We clearly have the following implications.

W_{α}(s)W_{α}(s)^{∗} ≥ W_{β}(s)W_{β}(s)^{∗}
⇒

W_{+}^{−L}(s)W_{α}(s)W_{α}(s)^{∗}W_{+}^{−L}(s)^{∗} ≥ W_{+}^{−L}(s)W_{β}(s)W_{β}(s)^{∗}W_{+}^{−L}(s)^{∗}

⇔

W_{+}^{−L}(s) ( W_{+}(s) 0 ) Q′′_{α}(s)^{−1}Q′′_{α}(s)^{−∗} ( \begin{matrix}W_{+}(s)^{∗} \\ 0 \end{matrix} ) W_{+}^{−L}(s)^{∗}

≥ W_{+}^{−L}(s) ( W_{+}(s) 0 ) Q′′_{β}(s)^{−1}Q′′_{β}(s)^{−∗} ( \begin{matrix}W_{+}(s)^{∗} \\ 0 \end{matrix} ) W_{+}^{−L}(s)^{∗}

⇔

( I 0 ) Q′′_{α}(s)^{−1}Q′′_{α}(s)^{−∗} ( \begin{matrix}I \\ 0 \end{matrix} ) ≥ ( I 0 ) Q′′_{β}(s)^{−1}Q′′_{β}(s)^{−∗} ( \begin{matrix}I \\ 0 \end{matrix} )

⇔
π_{1}Q′′_{α}(s)^{−1}Q′′_{α}(s)^{−∗}π_{1} ≥ π_{1}Q′′_{β}(s)^{−1}Q′′_{β}(s)^{−∗}π_{1}

Our only remaining task is to invert the first implication. To this end we show that W_{+}(s)W_{+}(s)^{−L}W_{α}(s) = W_{α}(s). Indeed, by Lemma 5.4, we have W_{+}(s)W_{+}(s)^{−L}W_{−}(s) = W_{−}(s). Therefore

\begin{aligned}W_{+}(s)W_{+}(s)^{−L}W_{α}(s) & = & W_{+}(s)W_{+}(s)^{−L}W_{−}^{e}(s)Q′_{α}(s) \\ & = & W_{−}^{e}(s)Q′_{α}(s) = W_{α}(s). \end{aligned}

Extending ( I 0 ) in a natural way to a projection proves the reverse implication.

(6) ⇔ (4) By our assumption, we have

∥gP_{H_{r}(Q′′_{α})}∥ ≥ ∥gP_{H_{r}(Q′′_{β})}∥,  ∀g ∈ H_{r}(Q_{+}).  (168)

We show now that this inequality holds for all g ∈ H_{+}^{2}π_{1}. Clearly, for all f ∈ H_{+}^{2}, we have f = fπ_{1} + fπ_{2}. In particular, since by equation (89) H_{r}(Q′′)π_{1} ⊂ H_{r}(Q_{+}^{e}), we have

H_{r}(Q′′) = H_{r}(Q′′)π_{1} + H_{r}(Q′′)π_{2} ⊂ H_{r}(Q_{+}^{e}) + H_{+}^{2}π_{2}.

So gP_{H_{r}(Q′′)} = gP_{H_{r}(Q′′)}(P_{H_{r}(Q_{+}^{e})} + P_{H_{+}^{2}π_{2}}). Assume g ∈ H_{+}^{2}π_{1}, i.e. g = fπ_{1}, then it follows that, for all f ∈ H_{+}^{2},

\begin{aligned}fπ_{1}P_{H_{r}(Q′′)} & = & fπ_{1}(P_{H_{r}(Q_{+}^{e})} + P_{H_{+}^{2}π_{2}})P_{H_{r}(Q′′)} \\ & = & fπ_{1}P_{H_{r}(Q_{+}^{e})}P_{H_{r}(Q′′)} \end{aligned}

This implies

∥fπ_{1}P_{H_{r}(Q′′_{α})}∥ = ∥fπ_{1}P_{H_{r}(Q_{+}^{e})}P_{H_{r}(Q′′_{α})}∥ ≥ ∥fπ_{1}P_{H_{r}(Q_{+}^{e})}P_{H_{r}(Q′′_{β})}∥ = ∥fπ_{1}P_{H_{r}(Q′′_{β})}∥.

Thus we have obtained

∥fπ_{1}P_{H_{r}(Q′′_{α})}∥ ≥ ∥fπ_{1}P_{H_{r}(Q′′_{β})}∥ ∀f ∈ H_{+}^{2}.  (169)

We take now special choices for f, namely

f = \frac{ξ}{s + \overline{ω}} , ξ ∈ Im π_{1}, Re ω > 0.

Note that for any H_{+}^{2} function h, we have ⟨h, \frac{ξ}{s + \overline{ω}} ⟩_{H_{+}^{2}} = h(ω)ξ^{∗}. It is easy to check that

\frac{ξ}{s + \overline{ω}} = \frac{ξ − ξQ(ω)^{∗}Q(s)}{s + \overline{ω}} + \frac{ξQ(ω)^{∗}Q(s)}{s + \overline{ω}}

is an orthogonal decomposition relative to H_{+}^{2} = H_{r}(Q) ⊕ H_{+}^{2}Q. In fact,

K(s, ω) := \frac{I − Q(ω)^{∗}Q(s)}{s + \overline{ω}}

is the reproducing kernel for H_{r}(Q). Inequality (169) translates therefore into

\frac{∥ξ∥^{2} − ξQ′′_{α}(ω)^{∗}Q′′_{α}(ω)ξ^{∗}}{ω + \overline{ω}} ≥ \frac{∥ξ∥^{2} − ξQ′′_{β}(ω)^{∗}Q′′_{β}(ω)ξ^{∗}}{ω + \overline{ω}} ,

or, since ω + \overline{ω} > 0

ξQ′′_{β}(ω)^{∗}Q′′_{β}(ω)ξ^{∗} ≥ ξQ′′_{α}(ω)^{∗}Q′′_{α}(ω)ξ^{∗}

This is equivalent to

π_{1}Q′′_{β}(ω)^{∗}Q′′_{β}(ω)π_{1} ≥ π_{1}Q′′_{α}(ω)^{∗}Q′′_{α}(ω)π_{1}.

Conversely, assume (163) holds. Computing backwards, it is easy to see that this entails

π_{1}K_{α}(ω, ω)π_{1} ≥ π_{1}K_{β}(ω, ω)π_{1}  (170)

where K_{i}(s, ω) := \frac{I−Q′′_{i}(ω)^{∗}Q′′_{i}(s)}{s+\overline{ω}}, for i = α, β.

We assume first that

H_{r}(Q′′_{α})π_{1} = H_{r}(Q′′_{β})π_{1}  (171)

and denote the common dimension of these spaces by r. Then we can choose in these spaces basis matrices F_{α} and F_{β}, which we consider as a column of r rows of H_{+}^{2}, such that F_{α}π_{1} = F_{β}π_{1} = F_{+} . This can be achieved by taking any basis F_{α} in H_{r}(Q′′_{α}) and constructing F_{β} as (F_{α}π_{1})(H_{r}(Q′′_{β})|π_{1})^{−1}, noting that, by Proposition 5.4, the map H_{r}(Q′′_{β})|π_{1} is injective. Then it is well known, see Dym [1989], that

K_{i}(s, ω) = F_{i}(ω)^{∗}P_{i}^{−1}F_{i}(s) i = α, β

where P_{i} is the Gram matrix of the basis F_{i} defined, for i = α, as (P_{α})_{j.k} = ⟨F_{j,α}, F_{k,α}⟩ where j, k = 1, ..., r and F_{j,α} is the j − th row of the matrix F_{α}. Therefore the relation (170) can be written as

π_{1}F_{α}(ω)^{∗}P_{α}^{−1}F_{α}(ω)π_{1} ≥ π_{1}F_{β}(ω)^{∗}P_{β}^{−1}F_{β}(ω)π_{1}

which, in view of our choice of the basis, becomes

F_{+}(ω)^{∗}P_{α}^{−1}F_{+}(ω) ≥ F_{+}(ω)^{∗}P_{β}^{−1}F_{+}(ω).  (172)

We claim that (172) entails

P_{α}^{−1} ≥ P_{β}^{−1}  (173)

To see this, we need to show that

F (ω)^{∗}P F (ω) ≥ 0  ω ∈ II  (174)

implies P ≥ 0. Assume û(ω) is a row vector in H_{+}^{2} of suitable dimension. Then denoting by F the Fourier transform, it is well known that

F^{−1}[F (ω)û(ω)](t) = ∫_{−∞}^{t} e^{A(t−s)}Bu(s)ds = x(t)

where u is the inverse transform of û and (A, B) denotes a controllable pair such that F (ω) = (ωI − A)^{−1}B. Similarly,

F^{−1}[û(ω)^{∗}F (ω)^{∗}P F_{+}(ω)û(ω)](t) = ∫_{−∞}^{t} x(s − t)^{∗}P x(s)ds

which evaluated at zero yields, taking into consideration inequality (174),

0 ≤ ∫_{−∞}^{0} x(s)^{∗}P x(s)ds  (175)

Suppose now ξ^{∗}P ξ < 0 for some ξ ∈ CI ^{m}. Then, since ξ^{∗}e^{A^{∗}s}P e^{As}ξ is continuous in s, there exists a τ < 0 such that the integral ∫_{τ}^{0} ξ^{∗}e^{A^{∗}s}P e^{As}ξds is strictly negative. On the other hand, in view of controllability, we can find a control u_{0} s.t. x(τ) = ∫_{−∞}^{τ} e^{A(τ−s)}Bu(s)ds = e^{−Aτ}ξ. Moreover, we can choose this control so that ∫_{−∞}^{τ} x^{∗}(s)P x(s)ds is arbitrarily small, in particular smaller than − ∫_{τ}^{0} ξ^{∗}e^{A^{∗}s}P e^{As}ξds. Furthermore, we set u_{0}(s) = 0 for τ ≤ s ≤ 0. Therefore we have constructed a control such that ∫_{−∞}^{0} x^{∗}(s)P x(s)ds < 0. As this contradicts (175), we can conclude that P ≥ 0.

If we go now back to (172), we immediately see that the above argument yields (173). To conclude, let now g ∈ H_{+}^{2}π_{1}. Then, by a well-known property of reproducing kernels, see Dym [1989],

gP_{H_{r}(Q′′_{i})} = ⟨g(ω), K(·, ω)⟩ = ⟨g, F_{i}⟩P_{i}^{−1}F_{i}

and

∥gP_{H_{r}(Q′′_{i})}∥^{2} = ⟨g, F_{i}⟩P_{i}^{−1}⟨F_{i}, g⟩

Since ⟨g, F_{α}⟩ = ⟨g, F_{β}⟩, we see that P_{α}^{−1} ≥ P_{β}^{−1} implies ∥gP_{H_{r}(Q′′_{α})}∥^{2} ≥ ∥gP_{H_{r}(Q′′_{β})}∥^{2}. This shows the implication under the restrictive assumption (171).

If the assumption (171) is not satisfied, then, necessarily, we must have H_{r}(Q′′_{α})π_{1} ⊃ H_{r}(Q′′_{β})π_{1}. Indeed, inequality (162) implies that

Q′′_{β}(−s̄_{j})π_{1}ξ_{j}^{∗} = 0 ⟹ Q′′_{α}(−s̄_{j})π_{1}ξ_{j}^{∗} = 0  (176)

and, more generally, if the superscript indicates the k-th derivative,

Q_{β}^{(k)}(−s̄_{j})π_{1}ξ_{j}^{∗} = 0 ⟹ Q_{α}^{(k)}(−s̄_{j})π_{1}ξ_{j}^{∗} = 0  (177)

Therefore, we conclude that if \frac{ξ}{(s−s_{j})^{k+1}} π_{1} ∈ H_{r}(Q′′_{β})π_{1} then \frac{ξ}{(s−s_{j})^{k+1}} π_{1} ∈ H_{r}(Q′′_{α})π_{1}, which is our claim. Thus we can decompose H_{r}(Q′′_{α})π_{1} as

H_{r}(Q′′_{α})π_{1} = H_{r}(Q′′_{β})π_{1} ⊕ [H_{r}(Q′′_{α})π_{1} ⊖ H_{r}(Q′′_{β})π_{1}]

This means that if F̂_{β} is a basis in H_{r}(Q′′_{β})π_{1}, we can find a basis F̂_{α} of H_{r}(Q′′_{α})π_{1} of the form

F̂_{α} = ( F̂_{β} ; x ); let now F_{i} be the inverse image in H_{r}(Q′′_{i}) under π_{1} of F̂_{i}, for i = α, β. Construct

the reproducing kernels K_{i}(s, ω) = F_{i}(ω)^{∗}P_{i}^{−1}F_{i}(s). Then

π_{1}K_{α}π_{1} = F̂_{α}(ω)^{∗}P_{α}^{−1}F̂_{α}(s)

and

π_{1}K_{β}π_{1} = F̂_{β}(ω)^{∗}P_{β}^{−1}F̂_{β}(s) = F̂_{α}(ω)^{∗} ( \begin{matrix}P_{β}^{−1} & 0 \\ 0 & 0 \end{matrix} ) F̂_{α}(s)

so that, again in view of the ordering on the matrices following from (172) we obtain P_{α}^{−1} ≥

( \begin{matrix}P_{β}^{−1} & 0 \\ 0 & 0 \end{matrix} ). The conclusion of the backward implication then follows as above and the equiv-

alence is thus proved.

(3) ⇔ (5) Assume first that

π_{1}\overline{H}_{c}((Q′′_{α})^{∗}) = π_{1}\overline{H}_{c}((Q′′_{β})^{∗}).  (178)

This is a coinvariant subspace of \overline{H}_{c}(Q_{+}^{∗}), hence of the form \overline{H}_{c}(Q^{∗}) for some left factor Q^{∗} of Q_{+}^{∗}. The common dimension of these subspaces will be denoted by r. Let \overline{F} be a basis matrix of \overline{H}_{c}(Q^{∗}). Since, by Proposition 5.4, the projections π_{1}|\overline{H}_{c}((Q′′_{i})^{∗}) are injective, we can find basis matrices \overline{F}_{α} and \overline{F}_{β} in \overline{H}_{c}((Q′′_{α})^{∗}) and \overline{H}_{c}((Q′′_{β})^{∗}) respectively such that \overline{F} = π_{1}\overline{F}_{α} = π_{1}\overline{F}_{β}. Let P_{α}, P_{β} be the Gram matrices of these bases. Thus the projection of g ∈ \overline{H}_{c}(Q_{+}^{∗}) on \overline{H}_{c}((Q′′_{i})^{∗}), for i = α, β, can be written as

P_{\overline{H}_{c}(Q_{i}^{−1})}g = \overline{F}_{i}P_{i}^{−1}⟨\overline{F}_{i}, g⟩ i = α, β,

and hence

∥P_{\overline{H}_{c}(Q_{i}^{−1})}g∥^{2} = ⟨g, \overline{F}_{i}⟩P_{i}^{−1}⟨\overline{F}_{i}, g⟩  i = α, β.  (179)

In view of our choice of \overline{F}_{i} it is, for all g ∈ π_{1}H_{−}^{2} and thus a fortiori for all g ∈ H_{c}(Q_{+}^{−1})

⟨g, \overline{F}_{α}⟩ = ⟨g, \overline{F}_{β}⟩ = ⟨g, \overline{F}_{+}⟩

Therefore, we can write:

∥P_{\overline{H}_{c}(Q_{i}^{−1})}∥^{2} = ⟨g, \overline{F}_{+}⟩P_{i}^{−1}⟨\overline{F}_{+}, g⟩ i = α, β

Since \overline{F} is a basis matrix, we have \{⟨g, \overline{F}_{i}⟩|g ∈ H_{c}(Q^{∗})\} = C^{r}. Therefore, we can conclude that

∥P_{\overline{H}_{c}((Q′′_{α})^{∗})}g∥ ≥ ∥P_{\overline{H}_{c}((Q′′_{β})^{∗})}g∥ ∀g ∈ \overline{H}_{c}(Q_{+}^{∗})

if and only if

P_{α}^{−1} ≥ P_{β}^{−1}.  (180)

We have shown that (164) is equivalent to (180). We show now that (180) is equivalent to (162). First, we show that if Q is inner in the right half plane and \overline{F} is a basis matrix for H_{c}(Q^{∗}) with Gram matrix P, then we have, for Re s > 0,

\frac{Q^{−1}(s)Q^{−1}(s)^{∗} − I}{s + \overline{s}} = \overline{F} (s)P^{−1}\overline{F} (s)^{∗}.  (181)

In fact, by a standard result (see Dym [1989]), the reproducing kernel of a coinvariant subspace H_{c}(Q) can be written as:

\frac{I − Q(s)Q(ω)^{∗}}{s + ω̄} = F (s)P^{−1}F (ω)^{∗},  (182)

where F is any basis matrix of H_{c}(Q) and P its Gram matrix

Let now \overline{F} = Q^{−1}F be a left coprime Douglas-Shapiro-Shields factorization of \overline{F}. Then \overline{F} is a basis matrix of \overline{H}_{c}(Q^{∗}) if and only if F is a basis matrix of H_{c}(Q). Therefore multiplying (182) on the left by Q^{−1}(s), on the right by Q^{−1}(ω)^{∗} and taking ω = s we get (181). We can now write the following chain of equivalences in Re s > 0.

π_{1}Q_{α}^{−1}(s)Q_{α}^{−1}(s)^{∗}π_{1} ≥ π_{1}Q_{β}^{−1}(s)Q_{β}^{−1}(s)^{∗}π_{1}
⇔
π_{1} \frac{Q_{α}^{−1}(s)Q_{α}^{−1}(s)^{∗}−I}{s+\overline{s}} π_{1} ≥ π_{1} \frac{Q_{β}^{−1}(s)Q_{β}^{−1}(s)^{∗}−I}{s+\overline{s}} π_{1}
⇔
π_{1}\overline{F}_{α}(s)P_{α}^{−1}\overline{F}_{α}(s)^{∗}π_{1} ≥ π_{1}\overline{F}_{β}(s)P_{β}^{−1}\overline{F}_{β}(s)^{∗}π_{1}
⇔
\overline{F}_{+}(s)P_{α}^{−1}\overline{F}_{+}(s)^{∗} ≥ \overline{F}_{+}(s)P_{β}^{−1}\overline{F}_{+}(s)^{∗}

The last line is clearly implied by P_{α}^{−1} ≥ P_{β}^{−1}. The opposite implication is seen as above. If the assumption (178) is not satisfied, we proceed as in the proof of the previous equivalence.

(6) ⇔ (5) Follows from the fact that the map J : H_{+}^{2} ⟶ H_{−}^{2} defined by

(gJ)(s) := g(−\overline{s})^{∗},

where H_{+}^{2} and H_{−}^{2} are row and column spaces respectively, is a unitary map and hence preserves ortogonality. In particular, given any inner function Q, we have H_{+}^{2} = H_{r}(Q) ⊕ H_{+}^{2}Q which, since (gQ)J = Q^{∗}(gJ), implies the direct sum decomposition H_{−}^{2} = H_{c}(Q^{∗})⊕Q^{∗}H_{−}^{2}. Moreover, we have for g ∈ H_{+}^{2},

(gP_{H_{r}(Q)})J = P_{\overline{H}_{c}(Q^{∗})}(gJ).

Finally, we note that H_{r}(Q_{+})J = \overline{H}_{c}(Q_{+}^{∗}), and this completes the proof of the claimed equivalence.

(8) ⇔ (6) We assume (167) holds. Since

H_{r}(K_{i})Q′′_{i} ⊕ H_{r}(Q′′_{i}) = H_{r}(K_{i}Q′′_{i}) = H_{r}(\overline{Q}′′_{i}K_{+}^{e}) = H_{r}(\overline{Q}′′_{i})K_{+}^{e} ⊕ H_{r}(K_{+}^{e}),

we have

∥f∥^{2} = ∥fP_{H_{r}(K_{i})Q′′_{i}}∥^{2} + ∥fP_{H_{r}(Q′′_{i})}∥^{2}, ∀f ∈ H_{r}(K_{+}^{e}).

Thus (167) holds if and only if

∥fP_{H_{r}(Q′′_{α})}∥ ≥ ∥fP_{H_{r}(Q′′_{β})}∥ ∀f ∈ H_{r}(K_{+}^{e}).  (183)

We claim that the last inequality implies

∥gP_{H_{r}(Q′′_{α})}∥ ≥ ∥gP_{H_{r}(Q′′_{β})}∥ ∀g ∈ H_{r}(Q_{+}^{e}).  (184)

To see this, note that the factorization Q′_{i}Q′′_{i} = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) implies the inclusion

H_{r}(Q′′_{i}) ⊂ H_{r}(Q_{+}^{e}) ⊕ H_{r}(R^{e}).  (185)

We also note that H_{r}(Q_{+}^{e}) ⊥ H_{r}(R^{e}). Therefore, using the inclusion (185), we have for f ∈ H_{r}(K_{+}^{e})

fP_{H_{r}(Q′′_{i})} = fP_{H_{r}( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} )}P_{H_{r}(Q′′_{i})}
= f[P_{H_{r}(Q_{+}^{e})} ⊕ P_{H_{r}(R^{e})}]P_{H_{r}(Q′′_{i})}
= fP_{H_{r}(Q_{+}^{e})}P_{H_{r}(Q′′_{i})}.

Now, by Corollary 4.2, we have H_{r}(K_{+}^{e})P_{H_{r}(Q_{+}^{e})} = H_{r}(Q_{+}^{e}) and hence (183) implies (184). Conversely, assume (165) holds. By (185), we have for every f ∈ H_{r}(K_{+}^{e})

fP_{H_{r}(Q′′_{i})} = fP_{H_{r}(Q_{+}^{e})⊕H_{r}(R^{e})}P_{H_{r}(Q′′_{i})}
= f[P_{H_{r}(Q_{+}^{e})} + P_{H_{r}(R^{e})}]P_{H_{r}(Q′′_{i})}
= fP_{H_{r}(Q_{+}^{e})}P_{H_{r}(Q′′_{i})},

as, clearly, H_{r}(K_{+}^{e})|P_{H_{r}(R^{e})} = 0. Now, inequality (165) implies, once again using Corollary 4.2, that for every f ∈ H_{r}(K_{+}^{e})

∥fP_{H_{r}(K_{1})Q′′_{1}}∥ = ∥fP_{H_{r}(Q_{+}^{e})}P_{H_{r}(K_{1})Q′′_{1}}∥ ≥ ∥fP_{H_{r}(Q_{+}^{e})}P_{H_{r}(K_{2})Q′′_{2}}∥ = ∥fP_{H_{r}(K_{2})Q′′_{2}}∥.

(6) ⇔ (7) Assume inequality (165) holds. This implies Ker (H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{α})}) ⊂ Ker (H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{β})}). Therefore there exists a map Y : H_{r}(Q′′_{α}) ⟶ H_{r}(Q′′_{β}) for which

H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{α})}Y = H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{β})},

i.e. the following diagram commutes

[figure: H_{r}(Q′′_{β}); H_{r}(Q′′_{α}); Y; H_{r}(Q_{+}^{e}); T_{β} = H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{β})}; T_{α} = H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{α})}]

Diagram 9

Such a map Y is unique if and only H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{α})} is surjective. This, by Proposition 5.4, is indeed the case. Inequality (165) shows that Y = T_{α}^{−1}T_{β} is contractive.

Conversely, assume there exists a unique contraction Y : H_{r}(Q′′_{α}) ⟶ H_{r}(Q′′_{β}) satisfying T_{α}Y = T_{β}. For any f ∈ H_{r}(Q′′_{α}) we have ∥fT_{α}^{−1}T_{β}∥ ≤ ∥f∥. Since H_{r}(Q_{+}^{e})|P_{H_{r}(Q′′_{α})} is surjective, we can write f = gP_{H_{r}(Q′′_{α})} = gT_{α}, with g ∈ H_{r}(Q_{+}). This implies (165).

(8) ⇔ (9) The proof is similar to the equivalence (6) ⇔ (7). Thus, assume (167) holds. This implies Ker H_{r}(K_{+}^{e})|P_{H_{r}(K_{β})Q′′_{β}} ⊂ Ker H_{r}(K_{+}^{e})|P_{H_{r}(K_{α})Q′′_{α}}. Hence there exists a map Z : H_{r}(K_{β})Q′′_{β} ⟶ H_{r}(K_{α})Q′′_{α} for which R_{β}Z = R_{α}. Z is uniquely determined if and only if R_{β} is surjective, which, by Proposition 6.1, is the case. Now from the inequality ∥fR_{α}∥ ≤ ∥fR_{β}∥, setting f = gR_{β}^{−1}, it follows that ∥gR_{β}^{−1}R_{α}∥ ≤ ∥f∥, i.e. Z is contractive. Conversely, assume there exists a unique contraction Z : H_{r}(K_{β})Q′′_{β} ⟶ H_{r}(K_{α})Q′′_{α} satisfying R_{β}Z = R_{α}. Thus, for all f ∈ H_{r}(K_{α})Q′′_{α}, we have ∥fR_{β}^{−1}R_{α}∥ ≤ ∥f∥. Setting f = gR_{β} we obtain ∥gR_{α}∥ ≤ ∥gR_{β}∥.

Remark: The equivalence between 1. and 8. is well known. For the uniform choice of basis in the state space formulation, see Caines and Delchamps [1980], and also Lindquist and Picci [1985] for the geometric approach from which these formulas are derived. The inequality (163) is very similar to the definition (155). Nevertheless, it is a more general statement about the partial ordering of inner functions, which coincides with the usual one when π_{1} = I. There is no need to introduce spectral factors to formulate this result. It seems to us that this is the actual key to the understanding of the order structure. We also point out that inequality (165) implies that the angle between the subspaces H_{r}(Q_{+}^{e}) and H_{r}(Q′′_{α}) is smaller than the angle between H_{r}(Q_{+}^{e}) and H_{r}(Q′′_{β}). Similarly inequality (167) implies that the angle between the subspaces H_{r}(K_{+}^{e}) and H_{r}(K_{β})Q′′_{β} is smaller than the angle between H_{r}(K_{+}^{e}) and H_{r}(K_{α})Q′′_{α}. These implications are one sided, but we refrain from a further discussion of this. Finally, one might expect that the equivalence of conditions 6. and 8. might be proved directly from the direct sum decompositions (127), however we failed to do so.

Corollary 6.1 With the previously defined partial order in W^{m}, W_{−}^{e} and W_{+}^{e} are the minimal and maximal elements respectively.

Proof: Clearly, in formula (165), the maximal element is Q_{+} and the minimal one is I, which are the inner functions corresponding to W_{+} and W_{−}, respectively.

This previous corollary explains the terminology of extremal factors for W_{±}^{e}, \overline{W} _{±}^{e}.

We turn now to the geometric structure of the set of minimal Markovian splitting subspaces. The reader familiar with stochastic realization theory will recognize the connection to the geometric constructs used in the stochastic domain. In particular, some of the spaces pertain to the zero dynamics associated with a given spectral factor, i.e. to the maximal output nulling inner (anti)-stabilizable subspaces and the input containing outer (anti)-detectable subspaces associated to any given minimal realization. This study was initialized in Lindquist, Michaletzky and Picci [1995]. A further analysis will be given in Gombani and Fuhrmann [1998] and Fuhrmann [1998].

To facilitate the reading we give here a table of the basic constructs used. At the risk of repeating ourselves, we point out again that the various spaces in the table are a result of a normalization, where with W_{+} we associate the state space H_{r}(K_{+}^{e}).

Space  W_{−}  W  W_{+}
X  H_{r}(K_{−})Q_{+}  H_{r}(K)Q′′  H_{r}(K_{+})

H^{−}  H_{+}^{2}π_{1}Q_{+}  H_{+}^{2}π_{1}Q_{+}  H_{+}^{2}π_{1}Q_{+}

H^{+}  H_{−}^{2}π_{1}K_{+}  H_{−}^{2}π_{1}K_{+}  H_{−}^{2}π_{1}K_{+}

S  H_{+}^{2}Q_{+}  H_{+}^{2}Q′′  H_{+}^{2}

\overline{S}  H_{−}^{2}K_{−}Q_{+}  H_{−}^{2}KQ′′  H_{−}^{2}K_{+}

X ∩ H^{−}  H_{r}(K_{−})Q_{+}  [H_{r}(K) ∩ H_{r}(K_{−})Q′]Q′′ H_{r}(K_{+}) ∩ H_{r}(K_{−})Q_{+}

dim(X ∩ H^{−})  δ(K_{−})  δ(K) − δ(Q′)  δ(K_{+}) − δ(Q_{+})

X ∩ H^{+}  H_{r}(K_{+}) ∩ H_{r}(K_{−})Q_{+} H_{r}(K)Q′′ ∩ H_{r}(K_{+})  H_{r}(K_{+})

dim(X ∩ H^{+})  δ(K_{−}) − δ(Q_{−})  δ(K) − δ(\overline{Q}′′)  δ(K_{+})

(H^{−})^{⊥}P_{X}  {0}  [H_{r}(Q′)P_{H_{r}(K)}]Q′′  H(Q_{+})P_{H_{r}(K_{+})}

dim((H^{−})^{⊥}P_{X})  0  δ(Q′)  δ(Q_{+})

(H^{+})^{⊥}P_{X}  H(Q_{−})K_{+}P_{H_{r}(K_{−})Q_{+}} (H_{r}(\overline{Q}′′)K_{+})P_{H_{r}(K)Q′′}  \{0\}

dim((H^{+})^{⊥})P_{X}  δ(Q_{−})  δ(\overline{Q}′′)  0

Table 10 Theorem 6.3 Let W be a minimal, stable spectral factor. Let W = \overline{W} K be its DSS factorization over H_{−}^{∞}. Let the inner functions Q′, Q′′, \overline{Q}′, \overline{Q}′′ be determined by Proposition 5.2, and let Q_{+}^{e}, Q_{−}^{e}, K_{+}^{e}, K_{−}^{e} be the inner functions for which diagram 2 commutes.

Define the spaces X, S and \overline{S} as in Proposition (6.1), i.e.

\begin{aligned}X & = & H_{r}(K)Q′′ \\ S & = & H_{+}^{2}Q′′ \\ \overline{S} & = & H_{−}^{2}KQ′′ \end{aligned}  (186)

and the subspaces H^{−} and H^{+} by

\begin{aligned}H^{−} & = & H_{+}^{2}π_{1}Q_{+} \\ H^{+} & = & H_{−}^{2}π_{1}K_{+} \end{aligned}  (187)

Then:

1. We have

\begin{aligned}X ∩ H^{−} & = & [H_{r}(K) ∩ H_{r}(K_{−})Q′]Q′′ \\ X ∩ H^{+} & = & H_{r}(K)Q′′ ∩ H_{r}(K_{+}^{e}) \end{aligned}  (188)

2. If X, Y ⊂ H_{+}^{2} and Y ⊂ X denote by Y_{X}^{⊥} the orthogonal complement of Y in X. We have

\begin{aligned}(X ∩ H^{−})_{X}^{⊥} & = & (H^{−})^{⊥}P_{X} = [H_{r}(Q′)P_{H_{r}(K)}]Q′′ \\ (X ∩ H^{+})_{X}^{⊥} & = & (H^{+})^{⊥}P_{X} = (H_{r}(\overline{Q}′′)K_{+}^{e})P_{H_{r}(K)Q′′} \end{aligned}  (189)

3. We have

\begin{aligned}dim(H^{+})^{⊥}P_{X} & = & δ(\overline{Q}′′) \\ dim(H^{−})^{⊥}P_{X} & = & δ(Q′) \end{aligned}  (190)

4. We have

\begin{aligned}dim(X ∩ H^{−}) & = & δ(K) − δ(Q′) \\ dim(X ∩ H^{+}) & = & δ(K) − δ(\overline{Q}′′) \end{aligned}  (191)

Proof:

1. In view of equation (75) and the fact, see Lemma 6.1, that Q_{+}π_{1} = π_{1}Q_{+}, we have the equality K_{−}^{e}Q_{+}^{e}π_{1} = \overline{Q}′KQ′′π_{1}. Thus we compute

X ∩ H^{−} = H_{r}(K)Q′′ ∩ H_{+}^{2}π_{1}Q_{+} = H_{r}(K)Q′′ ∩ H_{+}^{2}Q_{+}^{e}π_{1}
= \overline{H}_{r}(K^{∗})KQ′′ ∩ H_{+}^{2}(K_{−}^{e})^{∗}K_{−}^{e}Q_{+}^{e}π_{1}
= \overline{H}_{r}(K^{∗})KQ′′ ∩ [H_{+}^{2} + \overline{H}_{r}((K_{−}^{e})^{∗})]K_{−}^{e}Q_{+}^{e}π_{1}

= [H_{r}(K^{∗})(\overline{Q}′)^{∗} ∩ H_{+}^{2}(K_{−}^{e})^{∗}]K_{−}^{e}Q_{+}^{e}π_{1}
= [H_{r}(K^{∗})(\overline{Q}′)^{∗} ∩ (H_{+}^{2} ⊕ H_{r}((K_{−}^{e})^{∗}))]K_{−}^{e}Q_{+}^{e}π_{1}
= [H_{r}(K^{∗})(\overline{Q}′)^{∗} ∩ H_{r}((K_{−}^{e})^{∗})]K_{−}^{e}Q_{+}^{e}π_{1}
= H_{r}(K)Q′′ ∩ H_{r}(K_{−}^{e})π_{1}Q_{+}^{e}
= [H_{r}(K) ∩ H_{r}(K_{−}^{e})Q′]Q′′

where in the last equality we have used the fact that H_{r}(K_{−}^{e})π_{1} = H_{r}(K_{−}^{e}) and therefore the projection can be eliminated. Similarly

X ∩ H^{+} = H_{r}(K)Q′′ ∩ H_{−}^{2}π_{1}K_{+} = H_{r}(K)Q′′ ∩ H_{−}^{2}K_{+}^{e}
= H_{r}(K)Q′′ ∩ [H_{−}^{2} ⊕ H_{r}(K_{+})]π_{1}
= H_{r}(K)Q′′ ∩ H_{r}(K_{+}^{e}).

2. We compute first the orthogonal complement of X ∩H^{−} in L^{2}. We use the decomposition (H_{r}(K)Q′′)^{⊥} = H_{+}^{2}KQ′′ ⊕ H_{r}(Q′′) ⊕ H_{−}^{2} and (H_{+}^{2}Q_{+}^{e})^{⊥} = H_{−}^{2}Q_{+}^{e}, as well as the equality H_{−}^{2}Q_{+}^{e} = H_{−}^{2} ⊕ H_{r}(Q_{+}^{e}) to get

(X ∩ H^{−})^{⊥} = (H_{r}(K)Q′′ ∩ H_{r}(K_{−}^{e})Q_{+}^{e})^{⊥}
= (H_{r}(K)Q′′)^{⊥} + (H_{r}(K_{−}^{e})Q_{+}^{e})^{⊥}
= [H_{+}^{2}KQ′′ ⊕ H_{−}^{2}Q′′] + [H_{+}^{2}K_{−}^{e}Q_{+}^{e} ⊕ H_{−}^{2}Q_{+}^{e}]
= [H_{+}^{2}KQ′′ ⊕ H_{−}^{2}Q′′]
+ [H_{+}^{2}\overline{Q}′KQ′′ ⊕ H_{r}(R^{e}) ⊕ H_{−}^{2}Q_{+}^{e}]
= [H_{+}^{2}KQ′′ ⊕ H_{−}^{2}Q′′]
+ [H_{+}^{2}\overline{Q}′KQ′′ ⊕ H_{−}^{2} ⊕ H_{r}(Q′)Q′′ ⊕ H_{r}(Q′′)]

Projecting orthogonally on H_{r}(K)Q′′ we obtain the equality

(X ∩ H^{−})_{X}^{⊥} = H_{r}(Q′)Q′′P_{H_{r}(K)Q′′} = [H_{r}(Q′)P_{H_{r}(K)}]Q′′.

Similarly, we compute the L^{2} orthogonal complement of X ∩ H^{+}.

(X ∩ H^{+})^{⊥} = H_{+}^{2}KQ′′ + H_{r}(Q′′) + H_{−}^{2} + H_{+}^{2}K_{+} + H_{−}^{2}
= H_{+}^{2}KQ′′ + H_{r}(Q′′) + H_{−}^{2} + (H_{+}^{2}\overline{Q}′′ + H_{r}(\overline{Q}′′))K_{+}
= H_{+}^{2}KQ′′ + H_{r}(Q′′) + H_{−}^{2} + H_{+}^{2}\overline{Q}′′K_{+} + H_{r}(\overline{Q}′′)K_{+}
= H_{+}^{2}KQ′′ + H_{r}(Q′′) + H_{−}^{2} + H_{r}(\overline{Q}′′)K_{+}

Projecting orthogonally on X = H_{r}(K)Q′′ we get

(X ∩ H^{+})_{X}^{⊥} = (H_{r}(\overline{Q}′′)K_{+}^{e})P_{H_{r}(K)Q′′},

as all other components are orthogonal to X.

3. We need to show that dim(H_{r}(\overline{Q}′′)K_{+})P_{H_{r}(K)Q′′} = δ(\overline{Q}′′). For this it suffices to show that the map H_{r}(\overline{Q}′′)K_{+}|P_{H_{r}(K)Q′′} is injective. But this follows from Lemma 5.3. Similarly for H_{r}(Q′)Q′′|P_{H_{r}(K)Q′′} 4. It follows from (3), since dim(X ∩ H_{−}) = dim X − dim(X ∩ H_{−})_{X}^{⊥} = δ(K) − δ(Q′′).

Similarly for dim(X ∩ H_{−})

Remark: It should be noted that in the Lindquist-Picci framework the multiplicity of K_{+} could be strictly less than the rank of Φ. This implies that H^{−} is not necessarily contained in the past of the state process of the realization. Since we do assume that the multiplicity of K_{+} is m_{0}, this can never happen. As we said, this is a rather special situation and we will not dwell on it here.

The set W^{m_{0}} inherits the partial order from W^{m} and, as W_{−}^{e}, W_{+}^{e} ∈ W^{m_{0}}, they are the minimal and maximal elements respectively of W^{m_{0}} with respect to this partial order. The set

W^{m} and its relation to W^{m_{0}} will be studied via the arithmetic properties of inner functions.

Given a minimal, stable p × m spectral factor W, we say, again following Lindquist and Picci [1991], that a pair of internal spectral factors W_{0−}, W_{0+} ∈ W^{m_{0}} is a tightest internal bound for W if

1. We have

W_{0−} ≤ W ≤ W_{0+}.

2. Given W_{1}, W_{2} ∈ W^{m_{0}} for which W_{1} ≤ W ≤ W_{2} we have

\{ \begin{matrix}W_{1} & ≤ W_{0−} \\ W_{0+} & ≤ W_{2}. \end{matrix}

We can state now:

Proposition 6.3 . Let W be an arbitrary minimal, stable p × m spectral factor W. Then

1. A tightest internal bound for W exists.

2. The tightest internal bound can be characterized as follows. Let Q′, Q′′ be the inner functions determined by Proposition 5.2. Let

\{ \begin{matrix}Q′ & = & Q′_{1}Q′_{2} \\ Q′′ & = & Q′′_{2}Q′′_{1} \end{matrix}

be the internal-external and external internal factorizations of Q′ and Q′′ respectively. Then

\{ \begin{matrix}W_{0−} & = & W_{−}^{e}Q′_{1} \\ W_{0+} & = & W_{+}^{e}(Q′′_{1})^{∗}. \end{matrix}

Proof: Follows from Theorem 5.3.

Proposition 6.3 gave an arithmetic characterization of the tightest internal bound for a given minimal, stable spectral factor. We proceed next to derive a geometric characterization for it.

Proposition 6.4 Given a minimal, stable spectral factor W, let W_{0−}, W_{0+} be the tightest internal bound. Let S_{0−}, \overline{S}_{0−} and S_{0+}, \overline{S}_{0+} be defined, as in Proposition 6.1, for W_{0−} and W_{0+} respectively. Let H_{0} = L^{2}π_{1}, i.e. it is the subspace of L^{2} of all those functions whose last m − m_{0} coordinates vanish. Then, with the notation of Theorem 5.3, we have

1. We have S_{0−} = S ∩ H_{0}, i.e.

H_{+}^{2}QQ′′_{1} = H_{+}^{2}Q′′ ∩ L^{2}π_{1}.  (192)

2. We have S_{0+} = SP_{H_{0}}, i.e.

H_{+}^{2}π_{1}Q′′_{1} = H_{+}^{2}Q′′π_{1}.  (193)

3. We have \overline{S}_{0−} = \overline{S}P_{H_{0}}, i.e.

H_{−}^{2}K_{0−}QQ′′_{1} = H_{−}^{2}KQ′′π_{1}.  (194)

4. We have \overline{S}_{0+} = \overline{S} ∩ H_{0}, i.e.

H_{−}^{2}K_{0+}Q′′_{1} = H_{−}^{2}KQ′′ ∩ L^{2}π_{1}.  (195)

5. We have

S_{0−} ∩ \overline{S}_{0−} = H_{r}(K_{0−})QQ′′_{1},  (196)

and

S_{0+} ∩ \overline{S}_{0+} = H_{r}(K_{0+})Q′′_{1},  (197)

6. We have

X ∩ H_{0} = (H_{r}(K_{0+}) ∩ H_{+}^{2}Q)Q′′_{1},  (198)

and

dim X ∩ H_{0} = δ(K) − δ(Q′′_{2}).  (199)

Proof:

1. We proved in Proposition 5.4 that H_{r}(Q′′)π_{1} = H_{r}(QQ′′_{1}). This means that

H_{r}(Q′′)π_{1} + H_{+}^{2}π_{2} = H_{r}(QQ′′_{1}) + H_{+}^{2}π_{2}.

Taking orthogonal complements in H_{+}^{2}, we obtain

H_{+}^{2}Q′′ ∩ L^{2}π_{1} = H_{+}^{2}Q′′ ∩ H^{2}π_{1} = H_{r}(QQ′′_{1}).

2. We have

\begin{aligned}H_{+}^{2}Q′′π_{1} & = & H_{+}^{2}Q′′_{2}Q′′_{1}π_{1} \\ & = & H_{+}^{2}Q′′_{2}π_{1}Q′′_{1} ⊂ H_{+}^{2}π_{1}Q′′_{1}. \end{aligned}

We will show that H_{+}^{2}Q′′_{2}π_{1} = H_{+}^{2}π_{1}, which will prove the assertion. In fact, H_{+}^{2}Q′′_{2}π_{1} is an H_{+}^{∞}-invariant subspace of H_{+}^{2}π_{1}, hence of the form H_{+}^{2}π_{1}T for some inner function T.

Clearly, T has a representation of the form ( \begin{matrix}T_{0} & 0 \\ 0 & I \end{matrix} ) and is a right factor of Q′′_{2}, and so

also of Q′Q′′_{2}. Thus T Q′′_{1} is a right factor of Q_{+}, however this contradicts the fact that the greatest common right divisor of Q′′ and Q_{+}^{e} is Q′′_{1}. Thus T is necessarily constant unitary.

3. Referring to Diagram 4, we have the equality K_{0−}Q′_{2} = \overline{Q}′_{2}K. We compute

H_{−}^{2}KQ′′π_{1} = H_{−}^{2}(\overline{Q}′_{2})^{∗}K_{0−}Q′_{2}Q′′π_{1}
= H_{−}^{2}(\overline{Q}′_{2})^{∗}K_{0−}Q′_{2}Q′′_{2}Q′′_{1}π_{1}
= H_{−}^{2}(\overline{Q}′_{2})^{∗}K_{0−} ( \begin{matrix}Q & 0 \\ 0 & R \end{matrix} ) Q′′_{1}π_{1}
= H_{−}^{2}(\overline{Q}′_{2})^{∗}π_{1}K_{0−}QQ′′_{1}
= H_{−}^{2}π_{1}K_{0−}QQ′′_{1}.

We have used the fact that H_{−}^{2}(\overline{Q}′_{2})^{∗}π_{1} is an H_{−}^{∞}-invariant subspace of H_{−}^{2}, hence of the form H_{−}^{2}π_{1}T^{∗} for some inner function T. Such a function would be a common left factor of \overline{Q}′_{2}\overline{Q}′′ which is necessarily trivial.

4. Referring to Diagram 4, we have KQ′′_{2} = \overline{Q}′′_{2}K_{0+}. We compute

H_{−}^{2}KQ′′ ∩ L^{2}π_{1} = H_{−}^{2}KQ′′_{2}Q′′_{1} ∩ L^{2}π_{1}
= H_{−}^{2}\overline{Q}′′_{2}K_{0+}Q′′_{1} ∩ L^{2}π_{1}
= (H_{−}^{2}\overline{Q}′′_{2} ∩ L^{2}π_{1})K_{0+}Q′′_{1}
= H_{−}^{2}K_{0+}Q′′_{1}.

Using arguments as above, we used the fact that H_{−}^{2}\overline{Q}′′_{2} ∩ L^{2}π_{1} = H_{−}^{2} ∩ L^{2}π_{1}.

5. By trivial computations.

6. Since X = S ∩ \overline{S}, we have

X ∩ H_{0} = (S ∩ \overline{S}) ∩ H_{0} = (S ∩ H_{0}) ∩ (\overline{S} ∩ H_{0})
= S_{0−} ∩ S_{0+} = H_{+}^{2}QQ′′_{1} ∩ H_{−}^{2}K_{0+}Q′′_{1}
= (H_{r}(K_{0+}) ∩ H_{+}^{2}Q)Q′′_{1}.

By Proposition 5.6, the Toeplitz operator T_{K_{0+}Q^{∗}}^{r} is injective, hence, by Theorem 4.1, we have H_{+}^{2}K_{0+} ∩ H_{r}(Q) = 0. We compute

dim X ∩ H_{0} = dim H_{r}(K_{0+}) ∩ H_{+}^{2}Q = dim Ker T_{QK_{0+}^{∗}}^{r}
= codim (H_{+}^{2}K_{0+} + H_{r}(Q))
= δ(K_{0+}) − δ(Q)
= δ(K) − δ(Q′′_{2}).

This means that the dimension of X ∩ H_{0} is smaller than the dimension of X exactly by the number of external zeros of W.

Remarks. We would like to stress that the above result are much harder to prove if instead of the extended scattering pairs we try to use those defined in Proposition 6.2. A glance at the formulas appearing in Theorem 6.3 will convince the initiated reader that there are very close links here to geometric control theory. The study of these links is beyond the scope of this paper and will be relegated to subsequent publications.

We focus now on the analysis of the lattice structure of the sets W^{m} and X^{m}. Unhappily, we cannot prove that W^{m} is a complete lattice. In fact, in view of the fact that the set of all nonnegative matrices is not a complete lattice, see Ex. 9 on p. 142 of Halmos [1958], it is doubtful whether W^{m} is a complete lattice. However, some results are obtainable, in special cases. The analysis uses the arithmetic of inner functions. Of course, this arithmetic is reflected in the geometry of splitting subspaces. We begin by proving the following proposition. In following the proof, it may be convenient to refer to Diagram 4.

[figure: ^{\overline{W} {e \atop −}}; W_{−}^{e}; W_{+}^{e}; ^{\overline{W} {e \atop +}}; W_{β}; ^{\overline{W}^{β}}; W_{α}; ^{\overline{W}^{α}}; W_{∨}; ^{\overline{W}^{∨}}; W_{∧}; ^{\overline{W}^{∧}}; K_{−}^{e}; K_{+}^{e}; K_{∧}; K_{∨}; K_{β}; K_{α}]

Diagram 11

Proposition 6.5 Denote by W_{R}^{m} the set of all minimal, stable spectral factors of size p × m

for which, with the notation of Theorem 5.3, we have Q′Q′′ = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ).

Let W_{α}, W_{β} ∈ W_{R}^{m}. Let S_{W_{α}}, S_{W_{β}} be defined by (129). Then we have

1.

S_{W_{α}} ∩ S_{W_{β}} = H_{+}^{2}(Q′′_{α} ∨_{L} Q′′_{β}),  (200)

where Q′′_{α} ∨_{L} Q′′_{β} is the least common left inner multiple of Q′′_{α} and Q′′_{β}.

2.

S_{W_{α}} + S_{W_{β}} = H_{+}^{2}(Q′′_{α} ∧_{R} Q′′_{β}),  (201)

where Q′′_{α} ∧_{R} Q′′_{β} is the greatest common right inner divisor of Q′′_{α} and Q′′_{β}.

3.

\overline{S}_{W_{α}} ∩ \overline{S}_{W_{β}} = H_{−}^{2}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})K_{+}^{e}.  (202)

6 MINIMAL MARKOVIAN SPLITTING SUBSPACES 74

4.

\overline{S}_{W_{α}} + \overline{S}_{W_{β}} = H_{−}^{2}(\overline{Q}′′_{α} ∨_{L} \overline{Q}′′_{β})K_{+}^{e}  (203)

5. (a) There exists a unique inner function K_{∨} satisfying

K_{∨}(Q′′_{α} ∨_{L} Q′′_{β}) = (\overline{Q}′′_{α} ∨_{L} \overline{Q}′′_{β})K_{+}^{e}  (204)

(b) There exists a unique inner function K_{∧} satisfying

K_{∧}(Q′′_{α} ∧_{R} Q′′_{β}) = (\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})K_{+}^{e}  (205)

6. There exist inner functions Z, \overline{Z} for which

K_{∨}Z = \overline{Z}K_{∧}.  (206)

7. Define rational functions W_{∨}, W_{∧} by

\begin{aligned}W_{∧} & = & W_{+}^{e}(Q′′_{α} ∧_{R} Q′′_{β})^{∗} \\ W_{∨} & = & W_{+}^{e}(Q′′_{α} ∨_{L} Q′′_{β})^{∗}. \end{aligned}  (207)

Then W_{∨}, W_{∧} are minimal, stable spectral factors.

8. \overline{W}_{∨}, \overline{W}_{∧} defined by

\begin{aligned}\overline{W}_{∨} & = & W_{∨}K_{∨}^{∗} \\ \overline{W}_{∧} & = & W_{∧}K_{∧}^{∗}. \end{aligned}  (208)

are antistable spectral factors.

9. For Z, \overline{Z}, we have

\begin{aligned}W_{∨}Z & = & W_{∧} \\ \overline{W}_{∨}\overline{Z} & = & \overline{W}_{∧}. \end{aligned}  (209)

10. We have

(S_{W_{α}} ∩ S_{W_{β}}) ∩ (\overline{S}_{W_{α}} + \overline{S}_{W_{β}}) = H_{r}(K_{∨})(Q′′_{α} ∨_{L} Q′′_{β})  (210)

and

(S_{W_{α}} + S_{W_{β}}) ∩ (\overline{S}_{W_{α}} ∩ \overline{S}_{W_{β}}) = H_{r}(K_{∧})(Q′′_{α} ∧_{R} Q′′_{β}).  (211)

So (S_{W_{α}} ∩ S_{W_{β}}, \overline{S}_{W_{α}} + \overline{S}_{W_{β}}) is a scattering pair for W_{∨}.

Similarly, (S_{W_{α}} + S_{W_{β}}, \overline{S}_{W_{α}} ∩ \overline{S}_{W_{β}}) is a scattering pair for W_{∧}.

Proof:

1. We compute

S_{W_{α}} ∩ S_{W_{β}} = H_{+}^{2}Q′′_{α} ∩ H_{+}^{2}Q′′_{β} = H_{+}^{2}(Q′′_{α} ∨_{L} Q′′_{β}).

2. Similarly,

S_{W_{α}} + S_{W_{β}} = H_{+}^{2}Q′′_{α} + H_{+}^{2}Q′′_{β} = H_{+}^{2}(Q′′_{α} ∧_{R} Q′′_{β}).

3. The algebra becomes a bit more complicated in the complementary computations of \overline{S}_{W_{α}} ∩ \overline{S}_{W_{β}} and \overline{S}_{W_{α}} + \overline{S}_{W_{β}}. We have

\begin{aligned}K_{α}Q′′_{α} & = & \overline{Q}′′_{α}K_{+}^{e} \\ K_{β}Q′′_{β} & = & \overline{Q}′′_{β}K_{+}^{e} \end{aligned}

Therefore

\overline{S}_{W_{α}} ∩ \overline{S}_{W_{β}} = H_{−}^{2}K_{α}Q′′_{α} ∩ H_{−}^{2}K_{β}Q′′_{β} = H_{−}^{2}\overline{Q}′′_{α}K_{+}^{e} ∩ H_{−}^{2}\overline{Q}′′_{β}K_{+}^{e}
= [H_{−}^{2}\overline{Q}′′_{α} ∩ H_{−}^{2}\overline{Q}′′_{β}]K_{+}^{e} = [(H_{−}^{2} ⊕ H(\overline{Q}′′_{α})) ∩ (H_{−}^{2} ⊕ H(\overline{Q}′′_{β}))]K_{+}^{e}
= (H_{−}^{2} ⊕ (H(\overline{Q}′′_{α}) ∩ H(\overline{Q}′′_{β}))]K_{+}^{e} = (H_{−}^{2} ⊕ H(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})]K_{+}^{e}
= H_{−}^{2}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})K_{+}^{e}

4. Similarly,

\overline{S}_{W_{α}} + \overline{S}_{W_{β}} = H_{−}^{2}K_{α}Q′′_{α} + H_{−}^{2}K_{β}Q′′_{β} = H_{−}^{2}\overline{Q}′′_{α}K_{+}^{e} + H_{−}^{2}\overline{Q}′′_{β}K_{+}^{e}
= [H_{−}^{2}\overline{Q}′′_{α} + H_{−}^{2}\overline{Q}′′_{β}]K_{+}^{e} = [H_{−}^{2}(\overline{Q}′′_{α} ∨_{L} \overline{Q}′′_{β}]K_{+}^{e}
= H_{−}^{2}K_{∨}(Q′′_{α} ∨_{L} Q′′_{β}).

5. (a) By (69), we have

\begin{aligned}K_{α}Q′′_{α} & = & \overline{Q}′′_{α}K_{+}^{e} \\ K_{β}Q′′_{β} & = & \overline{Q}′′_{β}K_{+}^{e} \end{aligned}  (212)

Then we have

T = K_{α}Q′′_{α} ∨_{L} K_{β}Q′′_{β} = \overline{Q}′′_{α}K_{+}^{e} ∨_{L} \overline{Q}′′_{β}K_{+}^{e} = (\overline{Q}′′_{α} ∨_{L} \overline{Q}′′_{β})K_{+}^{e}.

Thus there exist inner functions Y_{α}, Y_{β} for which

T = Y_{α}Q′′_{α} = Y_{β}Q′′_{β}

Since Q′′_{α} ∨_{L} Q′′_{β} divides T on the right, it follows that there exists an inner function K_{∨} for which (204) holds.

(b) Similarly,

S = K_{α}Q′′_{α} ∧_{R} K_{β}Q′′_{β} = \overline{Q}′′_{α}K_{+}^{e} ∧_{R} \overline{Q}′′_{β}K_{+}^{e} = (\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})K_{+}^{e}.

Clearly, Q′′_{α} ∧_{R} Q′′_{β} divides S on the right, hence it follows that there exists an inner function K_{∧} for which (205) holds.

6. Clearly Q′′_{α} ∧_{R} Q′′_{β} divides Q′′_{α} ∨_{L} Q′′_{β} on the right. Thus there exists an inner function Z

satisfying

Q′′_{α} ∨_{L} Q′′_{β} = Z(Q′′_{α} ∧_{R} Q′′_{β}).  (213)

Analogously, there exists an inner function \overline{Z} satisfying

\overline{Q}′′_{α} ∨_{L} \overline{Q}′′_{β} = \overline{Z}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β}).  (214)

Next, we compute

K_{∨}Z(Q′′_{α} ∧_{R} Q′′_{β}) = K_{∨}(Q′′_{α} ∨_{L} Q′′_{β}) = (\overline{Q}′′_{α} ∨_{L} \overline{Q}′′_{β})K_{+}^{e}
= (\overline{Q}′′_{α} ∨_{L} \overline{Q}′′_{β})(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})^{∗}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})K_{+}^{e}
= \overline{Z}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})K_{+}^{e}
= \overline{Z}K_{∧}(Q′′_{α} ∧_{R} Q′′_{β}).

Hence equality (206) follows.

7. As (Q′′_{α} ∧_{R} Q′′_{β})^{∗} is unitary on the imaginary axis, it follows that W_{∧} is a spectral factor. The same holds for W_{∨}. We have W_{+}^{e} = W_{α}Q′′_{α} = W_{β}Q′′_{β}. This clearly implies that W_{∧} = W_{+}^{e}(Q′′_{α} ∧_{R} Q′′_{β})^{∗} ∈ H_{+}^{∞}. The minimality of W_{∧} follows from Theorem 5.1. Similarly, by our assumption that W_{α}, W_{β} ∈ W_{R}^{m}, we have that both Q′′_{α} and Q′′_{β} are right

factors of the inner function ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ). Hence Q′′_{α} ∨_{L} Q′′_{β} is also a right factor. This

implies W_{+}^{e}(Q′′_{α} ∨_{L} Q′′_{β})^{∗} ∈ H_{+}^{∞}. Invoking Theorem 5.1 once again, we conclude that W_{∨} is a minimal, stable spectral factor.

8. We compute

\overline{W}_{∧} = W_{∧}K_{∧}^{∗} = W_{+}^{e}(Q′′_{α} ∧_{R} Q′′_{β})^{∗}K_{∧}^{∗}
= W_{+}^{e}(K_{+}^{e})^{∗}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})^{∗}
= \overline{W} _{+}^{e}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})^{∗} ∈ H_{−}^{∞}

9. Using equality (213), we compute

\begin{aligned}W_{∨}Z & = & W_{+}^{e}(Q′′_{α} ∨_{L} Q′′_{β})^{∗}(Q′′_{α} ∨_{L} Q′′_{β})(Q′′_{α} ∧_{R} Q′′_{β})^{∗} \\ & = & W_{+}^{e}(Q′′_{α} ∧_{R} Q′′_{β})^{∗} = W_{∧}. \end{aligned}

Similarly, using equality (214), we compute

\overline{W}_{∨}\overline{Z} = W_{∨}K_{∨}^{∗}\overline{Z} = W_{∨}ZK_{∧}^{∗}
= W_{+}^{e}(Q′′_{α} ∨_{L} Q′′_{β})^{∗}(Q′′_{α} ∨_{L} Q′′_{β})(Q′′_{α} ∧_{R} Q′′_{β})^{∗}K_{∧}^{∗}
= W_{∧}K_{∧}^{∗} = \overline{W}_{∧}

10. We compute

(S_{W_{α}} ∩ S_{W_{β}}) ∩ (\overline{S}_{W_{α}} + \overline{S}_{W_{β}})
= H_{+}^{2}(Q′′_{α} ∨_{L} Q′′_{β}) ∩ H_{−}^{2}K_{∨}(Q′′_{α} ∨_{L} Q′′_{β})
= (H_{+}^{2} ∩ H_{−}^{2}K_{∨})(Q′′_{α} ∨_{L} Q′′_{β})
= H_{r}(K_{∨})(Q′′_{α} ∨_{L} Q′′_{β})

Similarly, we compute

(S_{W_{α}} + S_{W_{β}}) ∩ (\overline{S}_{W_{α}} ∩ \overline{S}_{W_{β}})
= H_{+}^{2}(Q′′_{α} ∧_{R} Q′′_{β}) ∩ H_{−}^{2}(\overline{Q}′′_{α} ∧_{R} \overline{Q}′′_{β})K_{+}^{e}
= H_{+}^{2}(Q′′_{α} ∧_{R} Q′′_{β}) ∩ H_{−}^{2}K_{∧}(Q′′_{α} ∧_{R} Q′′_{β})
= (H_{+}^{2} ∩ H_{−}^{2}K_{∧})(Q′′_{α} ∧_{R} Q′′_{β})
= H_{r}(K_{∧})(Q′′_{α} ∧_{R} Q′′_{β}).

Corollary 6.2 The set W_{0} of all minimal, stable, internal spectral factors is a complete lattice.

Proof: We apply the previous theorem for the case R = I.

The following is our version of Theorem 6.8 in Lindquist and Picci [1991].

Theorem 6.4 1. Let X_{i} ∈ X^{m_{0}}, i = 1, 2. Then the following statements are equivalent:

(a) X_{1} ≤ X_{2}.

(b) S_{1} ⊂ S_{2}.

(c) \overline{S}_{2} ⊂ \overline{S}_{1}.

2. Let X_{1} ∈ X^{m_{0}}, X_{2} ∈ X^{m}. Then the following statements are equivalent:

(a) X_{1} ≤ X_{2}.

(b) S_{1} ⊂ S_{2}.

(c) \overline{S}_{2}π_{1} ⊂ \overline{S}_{1}.

3. Let X_{2} ∈ X^{m_{0}}, X_{1} ∈ X^{m}. Then the following statements are equivalent:

(a) X_{1} ≤ X_{2}.

(b) S_{1}π_{1} ⊂ S_{2}.

(c) \overline{S}_{2} ⊂ \overline{S}_{1}.

Proof:

1. Assume X_{1} ≤ X_{2}. By Theorem 6.1, this is equivalent to W_{1}Q = W_{2} or to Q′′_{1} = QQ′′_{2}.

We compute

S_{W_{1}} = H_{+}^{2}Q′′_{1} = H_{+}^{2}QQ′′_{2} ⊂ H_{+}^{2}Q′′_{2} = S_{W_{2}}.

Conversely, assume S_{W_{1}} ⊂ S_{W_{2}}, i.e. H_{+}^{2}Q′′_{1} ⊂ H_{+}^{2}Q′′_{2}. This inclusion implies the existence of an inner function Q such that Q′′_{1} = QQ′′_{2}. Therefore

W_{2} = W_{+}^{e}(Q′′_{2})^{∗} = W_{+}^{e}(Q′′_{1})^{∗}Q = W_{1}Q,

that is W_{1} ≤ W_{2} and hence X_{1} ≤ X_{2}.

Assume W_{1} < W_{2}. Since both factors are internal, this means that there exists an inner function Q for which W_{2} = W_{1}Q. In turn we have, using the standard notation, that K_{1}Q = \overline{Q}K_{2}. We proceed to compute

\begin{aligned}\overline{S}_{2} & = & H_{−}^{2}K_{2}Q′′_{2} = H_{−}^{2}\overline{Q}^{∗}K_{1}QQ′′_{2} \\ & = & H_{−}^{2}\overline{Q}^{∗}K_{1}Q′′_{1} ⊂ H_{−}^{2}K_{1}Q′′_{1} = \overline{S}_{1}. \end{aligned}

Conversely, assume \overline{S}_{2} ⊂ \overline{S}_{1}, i.e. H_{−}^{2}K_{2}Q′′_{2} ⊂ H_{−}^{2}K_{1}Q′′_{1}. This means the existence of an inner function \overline{Q} for which \overline{Q}K_{2}Q′′_{2} = K_{1}Q′′_{1}. Since K_{i}Q′′_{i} = \overline{Q}′′_{i}K_{+}^{e}, we have \overline{Q}\overline{Q}′′_{2} = \overline{Q}′′_{1}. This means that \overline{W}_{1} ≤ \overline{W}_{2} and hence that W_{1} ≤ W_{2}. This is equivalent to X_{1} ≤ X_{2}.

2. Assume X_{1} ≤ X_{2} and X_{1} internal. Let X_{2−} be the greatest internal lower bound for X_{2}. By Proposition 6.4 we have S_{2−} = S_{2} ∩ H_{0}. Thus, by part 1,

S_{1} ⊂ S_{2−} = S_{2} ∩ H_{0} ⊂ S_{2}.

Conversely, assume S_{1} ⊂ S_{2}. This implies

S_{1} = S_{1} ∩ H_{0} ⊂ S_{2} ∩ H_{0} = S_{2−},

or X_{1} ≤ X_{2−} ≤ X_{2}.

Assuming X_{1} ≤ X_{2} we obtain as before X_{1} ≤ X_{2−}. Since X_{1}, X_{2−} are both internal, we can apply part 1, to get \overline{S}_{2−} ⊂ \overline{S}_{1}. Since, by Proposition 6.4, we have \overline{S}_{2−} = \overline{S}_{2}π_{1}, it follows that \overline{S}_{2}π_{1} ⊂ \overline{S}_{1}.

Conversely, assuming the last inclusion, we get X_{1} ≤ X_{2−} ≤ X_{2}.

3. This follows from the previous part by symmetry, or can be proved directly in an analogous way.

To get some feeling about external factors, we work out a very simple example.

Example: Let us take the minimum phase spectral factor W_{−} = \frac{z + 1}{z + 3} and the corresponding

maximum phase spectral factor W_{+} = \frac{1 − z}{z + 3}. Given a pair of complex numbers α, β, satisfying

|α|^{2} + |β|^{2} = 1, then

W = \frac{1}{z + 3} ( α(1 + z) β(1 − z) )  (215)

is a 1 × 2 spectral factor. We set

W_{−}^{e} = ( \frac{z + 1}{z + 3} 0 ) .

It is easily computed that

\begin{matrix}Q′ = ( \begin{matrix}α & β \frac{1 − z}{1 + z} \\ −\overline{β} & \overline{α}\frac{1 − z}{1 + z} \end{matrix} ) , Q′′ = ( \begin{matrix}\overline{α}\frac{1 − z}{1 + z} & −β \frac{1 − z}{1 + z} \\ \overline{β} & α \end{matrix} ) \\ Q′Q′′ = ( \begin{matrix}\frac{1 − z}{1 + z} & 0 \\ 0 & \frac{1 − z}{1 + z} \end{matrix} ) = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) \end{matrix}

6 MINIMAL MARKOVIAN SPLITTING SUBSPACES 79 For the DSS factorization we have

\overline{W} = \frac{1}{z − 3} ( α(1 + z) β(1 − z) )

and K = \frac{z − 3}{z + 3}. For f ∈ H_{r}(K_{+}^{e}), which up to a constant factor, is given by f = ( \frac{1}{z + 3} 0 ) we compute now,

fP_{H(Q′′)} = ( \frac{1}{z + 3} 0 ) (Q′′)^{∗}P_{−}Q′′
= ( \frac{1}{z + 3} 0 ) ( \begin{matrix}α\frac{1 + z}{1 − z} & β \\ −\overline{β} \frac{1 + z}{1 − z} & \overline{α} \end{matrix} ) P_{−}Q′′
= ( \frac{1 + z}{(z + 3)(1 − z)} β \frac{1}{z + 3} ) P_{−}Q′′
= ( \frac{1}{2} \frac{1}{1 − z} − \frac{1}{2} \frac{1}{z + 3} β \frac{1}{z + 3} ) P_{−}Q′′
= ( \frac{1}{2} \frac{1}{1 − z} 0 ) ( \begin{matrix}\overline{α}\frac{1 − z}{1 + z} & −β \frac{1 − z}{1 + z} \\ \overline{β} & α \end{matrix} )
= ( \frac{|α|^{2}}{2} \frac{1}{1 + z} \frac{−αβ}{2} \frac{1}{1 + z} )

This implies

∥fP_{H(Q′′)}∥^{2} = ∥f∥^{2} \{\frac{|α|^{4}}{4} + \frac{|α|^{2}|β|^{2}}{4} \} = ∥f∥^{2} \frac{|α|^{2}}{4} (|α|^{2} + |β|^{2}) = ∥f∥^{2} \frac{|α|^{2}}{4} .

Equivalently ∥fP_{H(Q′′)}∥ = ∥f∥\frac{|α|}{2}. The maximal value is obtained for |α| = 1.

Using this we compute

∥fP_{H(K_{−}^{e})Q_{+}^{e}}∥^{2} = ∥f∥^{2} − \frac{∥f∥^{2}}{4} = \frac{3∥f∥^{2}}{4} ,

or ∥fP_{H(K_{−}^{e})Q_{+}^{e}}∥ = \frac{\sqrt{3}∥f∥}{2}. Thus the angle between H(K_{−}^{e})Q_{+}^{e} and H_{r}(K_{+}^{e}) is, in this case, π/3.

Note that

lim_{|α|↦0} ∥H_{r}(K_{+}^{e})|P_{H_{r}(K)Q′′}∥ = 1,

but

lim_{|α|↦0} Q′′ ≠ I.

Next, we show that, except when α = 0, we have H(K_{+}^{e}) ∩ H_{+}^{2}Q′′ = {0}. Indeed, f ∈

H(K_{+}^{e}) ∩ H_{+}^{2}Q′′ if and only if f ∈ H(K_{+}^{e}) and f(Q′′)^{∗} ∈ H_{+}^{2}. Now ( \frac{1}{z + 3} 0 ) ∈ H_{+}^{2}Q′′ if

and only if

( \frac{1}{z + 3} 0 ) ( \begin{matrix}α\frac{1 + z}{1 − z} & β \\ −\overline{β} \frac{1 + z}{1 − z} & \overline{α} \end{matrix} ) = ( α\frac{1 + z}{(1 − z)(z + 3)} \frac{β}{z + 3} ) ∈ H_{+}^{2},

which is the case if and only if α = 0. Therefore, for all factors W of the form (215), we have H(K_{+}^{e}) ∩ H_{+}^{2}Q′′ = {0}.

7 The frame space

We turn now to the analysis of a space that contains all the zero and pole information of all spectral factors of given size. We define the frame space H_{m}^{□} of the spectral function Φ as the span of the set X^{m} of all minimal Markovian splitting subspaces associated with the set W^{m} of all minimal, stable p × m spectral factors.

We start by considering a special case.

Proposition 7.1 Assume Φ to be rank m_{0}. Let Q_{−}, Q_{+}, K_{−}, K_{+} be the inner functions as in Diagram 1. Then the frame space H_{m_{0}}^{□} is given by

H_{m_{0}}^{□} = H_{r}(K_{−}Q_{+}) = H_{r}(Q_{−}K_{+}).  (216)

Proof:

First, we compute

H_{r}(K)Q′′ ⊂ H_{r}(KQ′′) = H_{r}(\overline{Q}′′K_{+}) ⊂ H_{r}(\overline{Q}′\overline{Q}′′K_{+}) = H_{r}(Q_{−}K_{+}),

and hence we get the inclusion

⋁ H_{r}(K)Q′′ ⊂ H_{r}(Q_{−}K_{+}).  (217)

To prove the converse, we show that H_{r}(Q_{−}K_{+}) is spanned by the two Markovian splitting subspaces associated with W_{−} and W_{+}, namely H_{r}(K_{−})Q_{+} and H_{r}(K_{+}) respectively. From Corollary 4.2 it follows that the projection map H_{r}(Q_{+})|P_{H_{r}(K_{+})} is injective, which means that H_{+}^{2}K_{+} ∩ H_{r}(Q_{+}) = {0}. Applying Theorem 4.1, we have H_{+}^{2}Q_{+} + H_{r}(K_{+}) = H_{+}^{2}. Now, we have the following direct sum decomposition

H_{+}^{2}Q_{+} = [H_{+}^{2}K_{−} ⊕ H_{r}(K_{−})]Q_{+} = H_{+}^{2}K_{−}Q_{+} ⊕ H_{r}(K_{−})Q_{+}

and

H_{+}^{2} = H_{+}^{2}K_{−}Q_{+} ⊕ H_{r}(K_{−}Q_{+}).

Taken together, this implies H_{r}(K_{−})Q_{+} + H_{r}(K_{+}) = H_{r}(K_{−}Q_{+}), and so we get the inclusion

⋁ H_{r}(K)Q′′ ⊃ H_{r}(Q_{−}K_{+}).

The two inclusions imply the equality (216).

For the general case we will need the following lemma.

Lemma 7.1 Let R be the set of all k × k inner functions R having the same set of invariant inner factors r_{1}, …, r_{k}. We assume they are ordered so that r_{i}|r_{i−1}, and we set r_{1} = r. Then

⋁_{R∈R} H_{r}(R) = H_{r}(rI_{k}).  (218)

Proof: For each R ∈ R, the inner function r is its minimal inner function. This of course implies the inclusion H_{r}(R) ⊂ H_{r}(rI) and hence

⋁_{R∈R} H_{r}(R) ⊂ H_{r}(rI_{k}).

To see that this inclusion is actually an equality, it suffices to consider the set of all diagonal inner functions of the form R_{σ} = diag (r_{σ_{1}}, …, r_{σ_{k}}) with σ ∈ S_{k}, i.e. a permutation. Clearly we have R_{σ} ∈ R and

⋁_{σ∈S_{k}} H_{r}(R_{σ}) = H_{r}(rI_{k}),

and hence (218) follows.

We pass on now to the general case.

Theorem 7.1 Let Φ be a p × p, rank m_{0} spectral function. Let r be the scalar minimal inner function of Q_{+}, or equivalently of Q_{−}. Then the frame space H_{m}^{□} of Φ is given by

H_{m}^{□} = H_{r}(K_{−}Q_{+}) ⊕ H_{r}(rI_{m−m_{0}}).  (219)

Proof: First, we compute

H_{r}(K)Q′′ ⊂ H_{r}(KQ′′) = H_{r}(\overline{Q}′′K_{+}^{e})
⊂ H_{r}(\overline{Q}′\overline{Q}′′K_{+}^{e}) = H_{r} ( \begin{matrix}Q_{−}K_{+} & 0 \\ 0 & R \end{matrix} )
= H_{r}(Q_{−}K_{+}) ⊕ H_{r}(R) ⊂ H_{r}(Q_{−}K_{+}) ⊕ H_{r}(rI).

Thus ⋁ H_{r}(K)Q′′ ⊂ H_{r}(Q_{−}K_{+}) ⊕ H_{r}(rI), i.e.

H_{m}^{□} ⊂ H_{r}(Q_{−}K_{+}) ⊕ H_{r}(rI) = H_{m_{0}}^{□} ⊕ H_{r}(rI_{m−m_{0}}).

Obviously, we have H_{m_{0}}^{□} ⊕ {0} ⊂ H_{m}^{□}, so it suffices to show that ⋁ H_{r}(K)Q′′π_{2} = H_{r}(rI_{m−m_{0}}).

Choose any R satisfying R ≃ Q_{+}. Let Q′Q′′ = ( \begin{matrix}Q_{+} & 0 \\ 0 & R \end{matrix} ) be a balanced factorization which,

by Proposition 5.5, exists. Define W = W_{−}^{e}Q′ and let W = \overline{W} K be a DSS factorization. Now, by (103), we have H_{r}(K)Q′′π_{2} = H_{r}(R), and an application of Lemma 7.1 completes the proof.

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