ON THE CHARACTERIZATION AND PARAMETRIZATION OF MINIMAL

SPECTRAL FACTORS

P. A. Fuhrmann^{∗}^{†}

Department of Mathematics Ben-Gurion University of the Negev Beer Sheva, Israel

June 12, 1995

Abstract

A characterization of minimal spectral factors is given in terms of minimal factorizations of the phase function. The set of all minimal spectral factors is characterized via the set of solutions of a pair of algebraic Riccati equation.

∗Earl Katz Family Chair in Algebraic System Theory †Partially supported by the Israeli Academy of Sciences under Grant No. 249/90 and by GIF under Grant No. I 184.

1 Introduction

This paper focuses on the study of the set of all minimal spectral factors of a given spectral function Φ(s). Whlie this can be posed as a pure mathematical problem, much of the work on it stems from the work of Lindquist and Picci [1979,1985] on stochastic realization theory. In the work presented in this paper we will not restrict ourselves to the special case of rational spectral functions nor shall we aim at analysing the most general possible spectral functions. Our standing assumption will be that the spectral function is strictly noncyclic, see Fuhrmann [1981], and thus has essentially a meromorphic structure off the imaginary axis.

Of course, as soon as we depart from a finite dimensionality assumption, we have to reexamine our notion of the minimality of a spectral factor. We will depart from the current convention that defines the minimality of a spectral factor in terms of the existence of a corresponding Lindquist-Picci pair of spectral factors. Rather we will relate minimality to the existence of minimal factorizations of the phase function. This will lead as a consequence to the existence of a Lindquist-Picci pair for the corresponding spectral factor.

Once this correspondence is clarified we can characterize a minimal spectral factor in terms of the factorizations of two inner functions. Much more can be said if we restrict ourselves to the case of rational spectral functions. In that case, the factorizations of inner functions are related, by the results of Finesso and Picci [1982], to the solution of algebraic Riccati equations. Conbining this information with state space formulas derived for Lindquis-Picci pairs we are led to the complete parametrization of the set of all minimal spectral factors.

A recent effort has been made by Ferrante, Michaletzky and Pavon [1993], to substitute the analysis of a single algebraic Riccati equation, but of double the dimension, for the use of the two Riccati equations, as is done here. This can be done only by adding some mild restrictions on the spectral function. This approach uses an inner function rather than the phase function as we do in this paper. The use of the phase function has the clear advantage that the spectral function can be uniquely reconstructed from it. This is not the case with the inner function used by Ferrante, Michaletzky and Pavon.

The paper is structured as follows. In section 2 we introduce the class of strictly noncyclic functions and characterize them in terms of their Douglas, Shapiro and Shields, or DSS, factorizations. For this see Douglas, Shapiro and Shields [1971], and also Fuhrmann [1975] for the multivariable case. We follow this by the recalling the principal results on minimal factorizations. This is closely related to the series coupling of systems and follows Fuhrmann [1976,1981]. We conclude this section by stating the principal result on spectral factorizations that we will use in the rest of the paper.

We proceed, in section 3, to analyse four special spectral factors as well as several inner functions associated with them. We continue with the study the set of minimal spectral factors and their relation to Lindquist-Picci pairs.

Section 4 is devoted to a detailed study of rational inner functions and their factorizations. In this section we use both algebraic as well as analytic techniques. The use of both methods and perspectives prooves to be a very powerful combination. The connecting link is the fact that finite dimensional, backward invariant subspaces of H_{+}^{2} are identifiable with rational model spaces corresponding to stable, nonsingular polynomial matrices. Thus we can view invariant subspaces as related to factorization either of inner functions or alternatively to that of nonsingular polynomial matrices.

In section 5 we study the DSS factorization of proper, rational functions in L^{∞} from the state space point of view. The results generalize those obtained in Fuhrmann and Ober [1993] which were derived for H_{−}^{∞} functions only.

In section 6 we study, in the rational case, the various minimal, extremal spectral factors, introduced in section 3, in terms of zero and pole flipping. This exhibits clearly the connection with factorization theory and the solvability of Riccati equations. While the method is different, this section reproduces recent results of Picci and Pinzoni [1993].

In section 7 we summarize the state space formulas for the Lindquist-Picci and dual Lindquist-Picci pairs of an arbitrary, minimal spectral factor. Finally, in section 8, we have the main result of this paper, namely we give a state space parametrizartion of all minimal spectral factor. The free parameters are the nonnegative definite solutions to a pair of algebraic Riccati equations associated with the stable, minimum phase spectral factor.

The work in this paper was motivated by a visit to LADSEB and the Department of Electrical Engineering at the University of Padova. I am especially indebted to A. Ferrante, A. Gombani, M. Pavon, G. Picci and S. Pinzoni with whom I had numerous coversations. It is a pleasure to acknowledge this debt.

2 Strictly noncyclic functions

Let L^{∞} denote the space of measurable, essentially bounded functions on the imaginary axis. H_{+}^{2} will denote both the Hardy space of vector valued, analytic functions in the open right half plane which satisfy

∥f∥^{2} = sup_{x>0} ∫_{−∞}^{∞} ∥f(x + iy)∥^{2}dy < ∞,

and the subspace of L^{2} of their boundary values. Similarly we dfine H_{−}^{2}. As a cosequence of the Fourier-Plancherel and the Paley-Wiener theorems we have the orthogonal direct sum decomposition

L^{2} = H_{+}^{2} ⊕ H_{−}^{2}.

We denote by P_{+} and P_{−} the orthogonal projections of L^{2} onto H_{+}^{2} and H_{−}^{2} respectively. Given a function G ∈ L^{∞} the Hankel operator H_{G} : H^{2} ⟶ H_{−}^{2} is defined by

H_{G}f = P_{−}Gf f ∈ H_{+}^{2}.

We will denote by Ĥ_{Φ} : H_{−}^{2} ⟶ H_{+}^{2} the involuted Hankel operator defined by

Ĥ_{Φ}f = P_{+}(Φf), for f ∈ H_{−}^{2}.  (1)

Clearly we have, for Φ ∈ L^{∞}(iR),

(H_{Φ})^{∗} = Ĥ_{Φ^{∗}}  (2)

In the context of this paper an inner function is a matrix function in H_{+}^{∞} whose boundary values on the imaginary axis are unitary a.e. Thus, necessarily, an inner function is square.

In terms of inner functions, we can introduce the class of strictly noncyclic functions. The realization theory for this class of functions is studied in depth in Fuhrmann [1981].

Definition 2.1 1. A function G ∈ L^{∞} is called strictly noncyclic if there exists a square inner function \overline{M} ∈ H_{+}^{∞} such that

H_{−}^{2} ⊖ ImH_{G} = \{ImH_{G}\}^{⊥} = \overline{M}^{∗}H_{−}^{2}.  (3)

2. A function G ∈ L^{∞} is called doubly strictly noncyclic if both G and G^{∗} are strictly noncyclic.

We note that if a function G ∈ L^{∞} is strictly noncyclic and satisfies G = G^{∗} then it is doubly strictly noncyclic.

The following theorem is adapted from Fuhrmann [1975,1981,p.253] which, in turn, generalizes the scalar result derived previously in Douglas, Shapiro and Shields [1971].

Theorem 2.1 The following statements are equivalent.

1. G is doubly strictly noncyclic.

2. G is the strong radial limit of meromorphic functions of bounded type in both the open left and right half planes.

3. On the imaginary axis G admits factorizations

G = \overline{M}^{∗}\overline{N} = NM^{∗}  (4)

or equivalently

G = \overline{M}^{−1}\overline{N} = NM^{−1}  (5)

with all factors in H_{+}^{∞} and M, \overline{M} square inner. Moreover we can assume without loss of generality that the factorizations are coprime. Specifically that M, N are right coprime and \overline{M}, \overline{N} are left coprime.

We will refer to the factorizations (4) as the Douglas, Shapiro and Shields factorizations, or DSS factorizations of G.

All rational functions are strictly noncyclic. Of course G ∈ L^{∞} may be strictly noncyclic whereas G^{∗}, defined by G^{∗}(s) = G(−\overline{s})^{∗}, may fail to be so. However there are classes of functions which have this property. The simplest class of such functions other than rational functions are strictly noncyclic functions which satisfy the symmetry constraint relative to the imaginary axis, namely G^{∗} = G. Another class of interest is the class of strictly noncyclic all-pass functions, i.e. functions in L^{∞} that assume unitary values a.e. on the imaginary axis. In this case G^{∗} is also strictly noncyclic. Moreover, the numerator functions in the DSS factorization of G, N and \overline{N}, are also inner. This implies that for G^{∗} we have the DSS factorization

G^{∗} = \overline{N}^{∗}\overline{M} = MN^{∗}.  (6)

It is a consequence of Theorem 2.1.(ii) that the product of two strictly noncyclic functions is also strictly noncyclic. If we consider the product of two rational functions G_{1} and G_{2}, we know that the McMillan degree satisfies

δ(G_{2}G_{1}) ≤ δ(G_{2}) + δ(G_{1}).  (7)

Equality in (7) occurs if and only if there are no zero pole cancellations between the two factors. These conditions can be translated into coprimeness conditions for polynomially coprime factorizations of the G_{i}. For this we refer to Callier and Nahum [1975]. Thus in the rational case we say that a factorization of G = G_{2}G_{1} is minimal if

δ(G_{2}G_{1}) = δ(G_{2}) + δ(G_{1}).  (8)

To extend this notion of a minimal factorization to the nonrational case we have to generalize the notion of degree. Such a degree theory using determinants of inner functions was developed in Fuhrmann [1981]. Without elaborating on the details, this leads to the following definition.

Definition 2.2 Let G_{i}, = 1, 2 be two doubly strictly noncyclic functions and let G_{i} have the DSS factorizations

G_{i} = \overline{M}_{i}^{−1}\overline{N}_{i} = N_{i}M_{i}^{−1}  (9)

then the factorization

G = G_{2}G_{1}  (10)

is called minimal if

1. \overline{N}_{2}, \overline{M}_{1} are left coprime

2. M_{2}, N_{1} are right coprime.

This definition is motivated by the study of the series connection of linear systems as in Callier and Nahum [1975] and Fuhrmann [1976a]. The realization theory behind this study is given in full detail in Fuhrmann [1981].

In the sequel we will make use of the following technical proposition.

Proposition 2.1 Let U, V, T be matrix functions in H_{+}^{∞}, with T assumed nonsingular. Let V T^{−1}U ∈ H_{+}^{∞}.

1. Assume U and T are strongly left coprime. Then V = ZT for some Z ∈ H_{+}^{∞}.

2. Assume T is an inner function and U and T are left coprime. Then V = ZT for some Z ∈ H_{+}^{∞}.

2.1 Spectral factorizations

Spectral factorization theory has its origin in the work of Fejer [1915] who proved that a trigonometric polynomial p(e^{it}) which is nonnegative definite on the unit circle if and only if it can be written in the form p(e^{it}) = |r(e^{it})|^{2}, for some analytic polynomial r.

The body of work done on spectral factorization related problems is enormous. We refer to the survey paper Kailath [1974] for some additional references. Much of the pioneering work of Wiener was related to the area of prediction theory for stochastic processes and the construction of optimal filters. The spectral factorization technique is also of importance in the construction of optimal gains in the LQG optimal control problem. This has been reduced by Kalman [1960] to the solvability of Riccati equations. The algebraic Riccati equation has in general many solutions of which only one is stabilizing. The parametrization of the set of all solutions of an algebraic Riccati equation has been reduced in Willems [1971] to the analysis of the set of all invariant subspaces of a single operator. For more on this we refer to Shayman [1983], Finesso and Picci [1982] and Fuhrmann [1985]. An exposition of this connection between factorization theory and the analysis of invariant subspaces from the point of view of polynomial model theory is given in Fuhrmann [1989].

We will utilize the following theorem which is sufficient for our purposes.

Theorem 2.2 Let Φ ∈ L^{∞} be a coercive, i.e. |Φ| ≥ δ > 0, matrix function that is positive definite on the imaginary axis. Then Φ is strictly noncyclic if and only if there exists a strictly noncyclic matrix function G ∈ H_{+}^{∞} for which

Φ(it) = G(it)G(it)^{∗}  (11)

or equivalently

Φ(s) = G(s)G^{∗}(s)  (12)

where

G^{∗}(s) = G(−\overline{s})^{∗}.  (13)

Such a function G will be called a spectral factor of Φ. Note that in our definition of spectral factor there are no implicit zero or pole constraints.

Proof: Assume such a factorization exists and G is strictly noncyclic. Then for the involuted Hankel operators we have the obvious inclusion

ImĤ_{Φ} ⊂ ImĤ_{G}

and the strict noncyclicity of G implies that of Φ.

Conversely, assume Φ is strictly noncyclic. Since Φ is assumed to be coercive we have ∫_{−∞}^{∞} log ∥Φ∥\frac{dt}{1+t^{2}} < ∞. Hence, by standard results e.g. Helson [1964], a spectral factorization of the form (12) exists with G ∈ H_{+}^{∞}. Without loss of generality we may assume that G^{∗} is outer in H_{−}^{∞}. It remains to show that G is strictly noncyclic. To this end we check

ImĤ_{Φ} = {P_{+}GG^{∗}h|h ∈ H_{−}^{2}}

and as G^{∗} is outer the set {G^{∗}h|h ∈ H_{−}^{2}} is dense in H_{−}^{2}. Hence

ImĤ_{G} ⊂ \overline{ImĤ_{Φ}}.

and G is indeed strictly noncyclic.

Of course if Φ is assumed rational we would like to be more specific about the spectral factors. In fact we have the following resulti, see Youla [1961].

Theorem 2.3 Given a corcive, rational p×p spectral function Φ, then there exist a rational function G such that G, G^{−1} are both stable, i.e. in H_{+}^{∞} and

Φ = GG^{∗}  (14)

The rational spectral factorization result of Youla can be derived from a more specialized result, namely the generalization of Fejer’s theorem on the factorization of positive trigonometric polynomials. We prefer to state this result in its equivalent half plane setting. This result has a fairly long history. The first rigorous proof seems to be to Coppel [1974a,b], although there have been previous attempts, especially by Jakubovic [1970]. Coppel’s proof is reproduced and generalized in Gohberg, Lancaster and Rodman [1982]. Other proofs have been given by Callier [1985], Dym and Young [199?], to quote a few.

Theorem 2.4 Let Q(s) be a polynomial matrix that satisfies Q^{∗} = Q(s) and is nonnegative definite on the imaginary axis. Then there exists a stable polynomial matrix P such that

Q(s) = P (s)P^{∗}(s).  (15)

Throughout the rest of this paper we will assume that the spectral function is strictly positive, or coercive, i.e. there exists a δ > 0 for which

Φ(it) > δI  (16)

on the imaginary axis including the point at infinity.

We proceed now with the analysis of some special spectral factors.

Theorem 2.5 Let Φ be a coercive, strictly noncyclic spectral function. Then there exists a spectral factor W_{−} which is W_{−}, W_{−}^{−1} ∈ H_{+}^{∞}.

Proof: Let W be a stable spectral factor whose existence is guarranteed by Theorem 2.2. Let W = W_{−}S be an outer/inner factorization of W (obtained from an inner/outer factorization of W̃). Since S is inner it is clear from Φ = W W^{∗} = W SS^{∗}W^{∗} = W_{−}W_{−}^{∗} that W_{−} is also a stable spectral factor. Moreover, as W_{−} is outer, W_{−}^{−1} is analytic in the open right half plane. It is actually in H_{+}^{∞} as a consequence of the coercivity of Φ and the maximum principle.

We will refer to W_{−} as the stable, minimum phase spectral factor. In the rational setting this means that all the zeroes and poles of W_{−} are located in the open right half plane.

The existence, for a coercive spectral function Φ, of an essentially unique, stable, minimum phase spectral factor, leads also to the existence of an essentially unique, antistable, maximum phase (i.e. conjugate outer) spectral factor.

Theorem 2.6 Let Φ be a coercive spectral function. Then there exists a unique, up to a constant unitary factor, spectral factor \overline{W}_{+} which is antistable and maximum phase.

Proof: Let us consider the spectral function Φ^{−1}. By Theorem 2.5 there exists an essentially unique, stable, minimum phase spectral factor V_{−}. That is Φ^{−1} = V_{−}V_{−}^{∗}. This implies Φ = V_{−}^{−∗}V_{−}^{∗}, or

Φ = \overline{W}_{+}\overline{W} _{+}^{∗}

where we define \overline{W}_{+} = V_{−}^{−∗}. Clearly \overline{W}_{+} is antistable and maximum phase.

We recall that an all-pass function is a function that assumes unitary values on the imaginary axis. This allows us to state the following simple yet important proposition.

Proposition 2.2 Let W_{1}, W_{2} be any two spectral factors of Φ. Then W_{1}^{−1}W_{2} is an all pass functions.

Proof: We have

Φ = W_{1}W_{1}^{∗} = W_{2}W_{2}^{∗}  (17)

By coercivity, both W_{1} and W_{2} are invertible elements of L^{∞}. Hence

W_{1}^{−1}W_{2}W_{2}^{∗}W_{1}^{−∗} = I

and so W_{1}^{−1}W_{2} is an all-pass function.

The all-pass functions that arise out of Proposition 2.2 have interesting properties if the spectral factors W_{1}, W_{2} are appropriately chosen. This will be explored in greater detail in the next section.

3 Spectral factors

In this section we study and characterize the extremal, minimal spectral factors. Moreover we give a characterization, in these terms, for all minimal stable, or minimum phase, spectral factors. We introduce the phase function corresponding to the spectral function Φ. Minimal spectral factorizations are defined in terms of minimal factorizations of the phase function.

Next we introduce the concept of a Lindquist-Picci pair and give a characterization of the existence of minimal spectral factors in terms of the existence of a Lindquist-Picci pair. We follow this by relating Lindquist-Picci pairs to DSS factorizations. This is of great importance for the computation of state space formulas.

The following proposition establishes the existence of two special spectral factors and characterizes their minimality properties. It is a direct consequence of the existence of innerouter factorizations.

Proposition 3.1 1. Let W_{−} be the outer spectral factor of a coercive spectral function Φ. Then W is a stable spectral factor of Φ if and only if W = W_{−}Q for some inner function Q.

2. Let \overline{W}_{+} be the conjugate outer spectral factor of Φ. Then \overline{W} is an antistable spectral factor of Φ if and only if \overline{W} = \overline{W}_{+}Q^{∗} for some inner function Q.

Proof:

1. Follows directly from the existence of inner-outer factorizations.

2. Given an antistable spectral factor \overline{W} of Φ, then \overline{W}^{−∗} is a stable spectral factor of Φ^{−1}. Thus there exists an inner function Q for which \overline{W}^{−∗} = \overline{W} _{+}^{−1}Q. This is equivalent to the statement.

We proceed to construct out of the two spectral factors W_{−}, \overline{W}_{+} two additional, extremal spectral factors W_{+}, \overline{W}_{−} and characterize their minimality properties.

Theorem 3.1 Let Φ be a coercive, strictly noncyclic spectral function. Let W_{−}, \overline{W}_{+} be as above. Then

1. There exists an essentially unique spectral factor W_{+} which is stable and maximum phase and such that a function W is a stable, maximum phase spectral factor if and only if W = W_{+}R, for some inner function R.

2. There exists an essentially unique spectral factor \overline{W}_{−} which is antistable and minimum phase and such that a function W is a antistable, minimum phase spectral factor W if and only if W = \overline{W}_{−}R^{∗}, for some inner function R.

Proof:

1. Let

J = \{P ∈ H_{+}^{∞}|\overline{W}_{+}P ∈ H_{+}^{∞}\}.

As \overline{W}_{+} is strictly noncyclic, J is a nonempty, w^{∗} closed, left ideal in H_{+}^{∞}, thus it has, see Fuhrmann [1981], the Beurling type representation J = K_{+}H_{+}^{∞} for some inner function K_{+}. So we have

W_{+} = \overline{W}_{+}K_{+}.  (18)

Since W_{+} is stable, it follows from Proposition 3.1 that there exists an inner function Q_{+} such that

W_{+} = W_{−}Q_{+}.  (19)

By the minimality property of K_{+}, it follows that necessarily K_{+}, Q_{+} are right coprime.

Assume R is inner and let W = W_{+}R. Obviously W is stable and maximum phase as both W_{+} and R are such.

Conversely, let W be any stable, maximum phase spectral factor. By Lemma 3.1 we have W = W_{−}Q for some inner function Q. On the other hand we also have W_{+} = W_{−}Q_{+} and by the minimality of Q_{+}, necessarily W_{+} and Q_{+} are right coprime. We can write therefore

W = W_{−}Q = W_{+}Q_{+}^{−1}Q ∈ H_{+}^{∞}.

Applying Proposition 2.1 we have Q_{+}^{−1}Q = R ∈ H_{+}^{∞}. Since Q_{+}^{−1}Q is all pass, R is necessarily inner. So Q = Q_{+}R and hence W = W_{−}Q = (W_{−}Q_{+})R = W_{+}R.

2. Similarly, there exists an essentially unique inner function K_{−} for which \overline{W}_{−} = W_{−}K_{−}^{∗} ∈

H_{−}^{∞}. Equivalently

\overline{W}_{−} = \overline{W}_{+}K_{−}.  (20)

By Theorem 3.1, there exists an inner function Q_{−} for which \overline{W}_{−} = \overline{W}_{+}Q_{−}^{∗}, or, equivalently,

\overline{W}_{+} = \overline{W}_{−}Q_{−}.  (21)

It is convenient to consolidate the four equations (18)-(21) into the following diagram.

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

In the rational case all these spectral factors are connected via pole and zero flipping between the two principal half planes. We will go into the details of this in section 6.

Next we introduce the all important phase function through the following.

Definition 3.1 Let Φ be a coercive spectral function. The phase function T_{0} corresponding to Φ is defined by

T_{0} = \overline{W} _{+}^{−1}W_{−}  (22)

where W_{−} and \overline{W}_{+} are the stable, minimum phase and antistable, maximum phase spectral factors respectively.

The phase function is a non inner all-pass function. There is another, closely related, inner function defined by

U_{M} = \overline{W} _{−}^{−1}W_{+}.

The inner function U_{M} plays the central role in the Ferrante, Michaletzky and Pavon [1993] approach to the parametrization problem.

Corollary 3.1 The phase function T_{0} has the following DSS coprime factorizations

T_{0} = K_{+}Q_{+}^{∗} = Q_{−}^{∗}K_{−}.  (23)

Definition 3.2 A spectral factor W_{1} of Φ is called a minimal spectral factor If the following factorization of the phase function T_{0}

T_{0} = \overline{W} _{+}^{−1}W_{−} = (\overline{W} _{+}^{−1}W_{1})(W_{1}^{−1}W_{−})  (24)

is a minimal factorization.

Theorem 3.2 1. A spectral factor W is a minimal, stable spectral factor if and only if W = W_{−}Q for some left inner factor of Q_{+}.

2. \overline{W} is a minimal, antistable factor if and only if \overline{W} = \overline{W}_{+}Q^{∗} for some right inner factor Q of Q_{−}.

Proof:

1. Assume W = W_{−}Q and Q a left inner factor of Q_{+}, i.e. Q_{+} = QR. Then

T_{0} = \overline{W} _{+}^{−1}W_{−} = \overline{W} _{+}^{−1}W W^{−1}W_{−} = \overline{W} _{+}^{−1}W Q^{∗}

or, since T_{0} = K_{+}Q_{+}^{∗},

\overline{W} _{+}^{−1}W_{−} = K_{+}Q_{+}^{∗} = K_{+}R^{∗}Q^{∗} = (K_{+}R^{∗})Q_{+}^{∗}.

Obviously T_{0} = (K_{+}R^{∗})Q^{∗} is a minimal factorization as R^{∗}Q^{∗} = Q_{+}^{∗}, and K_{+} and Q_{+} are right coprime.

Conversely, assume W is a minimal, stable spectral factor. In this case we have W = W_{−}Q for some inner function Q. So it follows that

T_{0} = \overline{W} _{+}^{−1}W W^{−1}W_{−} = \overline{W} _{+}^{−1}W Q^{∗}

Now

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

Since W_{+}^{−1}W is an all pass function, we can write

W_{+}^{−1}W = RS^{∗} = \overline{S}^{∗}\overline{R}

with R, S right coprime and \overline{R}, \overline{S} left coprime. Hence T_{0} = (K_{+}RS^{∗})Q^{∗}. But T_{0} = K_{+}Q_{+}^{∗} and so RS^{∗}Q^{∗} = Q_{+}^{∗}. Thus the factorization

T_{0} = K_{+}Q_{+}^{∗} = K_{+}RS^{∗}Q^{∗}

is a minimal factorization if and only if R is a constant unitary matrix. Redefining S we have Q_{+} = QS and Q is a left inner factor of Q_{+}.

2. Follows by duality considerations.

We define now the notion of a Lindquist-Picci pair.

Definition 3.3 1. A pair \{W, \overline{W} \} of (minimal) spectral factors, with W stable and \overline{W} antistable, is called a Lindquist-Picci pair if there exist inner functions \overline{Q}_{2}, Q_{1} such that (a) W = W_{−}Q_{1}.

(b) \overline{W}_{+} = \overline{W} \overline{Q}_{2}.

(c) K = \overline{W}^{−1}W is inner, with K, \overline{Q}_{2} left coprime and K, Q_{1} right coprime.

2. Let W_{1} be a spectral factor of a coercive spectral function Φ. A pair \{W, \overline{W} \} is a Lindquist-Picci pair for W_{1} if it is a Lindquist-Picci pair and the functions U, V defined by

\begin{aligned}V & = & W_{1}^{−1}W \\ U & = & \overline{W}^{−1}W_{1} \end{aligned}  (25)

are both inner.

[figure: ^{\overline{W}^{−}}; ^{\overline{W}^{+}}; W_{+}; W_{−}; W; W_{1}; \overline{W}; K_{−}; U; V; K_{+}; A; B]

Given a minimal spectral factor W_{1}, its Lindquist-Picci pair gives the extremal minimal spectral factors, stable and antistable, that have the same zero structure. A similar situation occurs if we preserve the pole structure and let the zero structure vary from minimum phase to maximum phase. This leads to the following definition.

Definition 3.4 1. A pair \{W_{δ}, \overline{W}_{δ}\} of (minimal) spectral factors, with W_{δ} minimum phase and \overline{W}_{δ} maximum phase, is called a dual Lindquist-Picci pair if there exist inner functions \overline{K}_{1}, K_{2} such that

(a) W_{−} = W_{δ}K_{2}.

(b) \overline{W}_{δ} = \overline{W}_{+}\overline{K}_{1}.

(c) Q = W_{δ}^{−1}\overline{W}_{δ} is inner, with Q, K_{2} left coprime and Q, \overline{K}_{1} right coprime.

2. Let W_{1} be a spectral factor of a coercive spectral function Φ. A pair \{W_{δ}, \overline{W}_{δ}\} is a dual Lindquist-Picci pair for W_{1} if it is a dual Lindquist-Picci pair and the functions \overline{U}, \overline{V} defined by

\begin{aligned}\overline{U} & = & W_{δ}^{−1}W_{1} \\ \overline{V} & = & W_{1}^{−1}\overline{W}_{δ} \end{aligned}  (26)

are both inner.

As a result of our definition of minimality of a spectral factor we get the following theorem, giving a characterization of minimality in terms Lindquist-Picci pairs. We wish to note that it is also possible to develop the theory in which minimality is defined in terms of Lindquist-Picci pairs.

Theorem 3.3 1. A spectral factor W_{1} of a coercive spectral function Φ is minimal if and only if there exists a LP pair (W, \overline{W}) for W_{1}.

2. A spectral factor W_{1} of a coercive spectral function Φ is minimal if and only if there exists a dual LP pair (W_{δ}, \overline{W}_{δ}) for W_{1}.

Proof:

1. Assume there exists a Lindquist Picci pair for W_{1}. Considering the following diagram

[figure: ^{\overline{W}^{−}}; ^{\overline{W}^{+}}; W_{+}; W_{−}; W; W_{1}; \overline{W}; K_{−}; U; V; K_{+}; ^{\overline{Q}^{2}}; Q_{1}] we get

T_{0} = (\overline{Q}_{2}^{∗}U)(V Q_{1}^{∗}) = \overline{Q}_{2}^{∗}KQ_{1}^{∗}

with K, \overline{Q}_{2} left coprime and K, Q_{1} right coprime. Thus the above factorization is a minimal factorization and hence W_{1} is a minimal factor.

Conversely, assume W_{1} is a minimal spectral factor. Thus

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

is a minimal factorization. Let \overline{W} _{+}^{−1}W_{1} = \overline{Q}_{2}^{∗}U with \overline{Q}_{2}, U left coprime, and W_{1}^{−1}W_{−} = V Q_{1}^{∗} with V, Q_{1} right coprime. Set K = UV. By the assumption of minimality we have that K, \overline{Q}_{2} are left coprime and K, Q_{1} are right coprime. Defining \overline{W} = \overline{W}_{+}\overline{Q}_{2}^{∗} and W = W_{−}Q_{1} then W is stable and \overline{W} antistable. Moreover the pair \{W, \overline{W} \} is a Lindquist-Picci pair for W_{1}.

2. Follows by duality considerations.

It should be noted that the concepts of primal and dual Lindquist-Picci pairs are very closely related to Douglas, Shapiro and Shields factorizations. In fact, as we shall show explicitly in section 7, the computation of a Lindquist-Picci pair of a minimal spectral factor W_{1} is the same as the computation of its DSS factorizations. we have the following.

Proposition 3.2 Let W_{1} be a minimal spectral factor of a coercive, rational spectral function Φ. Let

\{ \begin{matrix}W_{1} & = & N_{1}M_{1}^{−1} \\ W_{1}^{∗} & = & M_{2}^{−1}N_{2} \\ W_{1}^{−1} & = & M_{3}^{−1}N_{3} \\ W_{1}^{−∗} & = & N_{4}M_{4}^{−1} \end{matrix}  (27)

be DSS factorization with M_{i} normalized inner functions. Then

1. {N_{1}, N_{2}^{∗}} is a Lindquist-Picci pair for W_{1}.

2. {N_{3}^{−1}, N_{4}^{−∗}} is a dual Lindquist-Picci pair for W_{1}.

Proof: Clearly W_{1}M_{1} = N_{1}, i.e. M_{1} stabilizes W_{1} by right multiplication and, using the right coprineness of N_{1} and M_{1}, it is the minimal inner function to do so.

The other parts are proved using similar reasoning.

Next we explore the connection between factorizations of the inner functions Q_{+} and K_{−}, or alternatively of Q_{−} and K_{+}, and LP pairs.

Proposition 3.3 1. A factorization Q_{+} = Q_{1}Q_{2} determines a unique LP pair and, conversely a LP pair determines a unique factorization of Q_{+}.

2. A factorization K_{−} = K_{1}K_{2} determines a unique dual LP pair and, conversely a dual LP pair determines a unique factorization of K_{−}.

Proof:

1. Let Q_{+} = Q_{1}Q_{2}. We define W = W_{−}Q_{1}. So, by Theorem 3.2, W is a minimal, stable spectral factor. Let K be the minimal inner function for which W K^{∗} is antistable. By Theorem 3.2, \overline{W} = \overline{W}_{+}\overline{Q}_{2}^{∗} for some inner function \overline{Q}_{2}. Clearly, by the minimality property of K, necessarily K and \overline{Q}_{2} are left coprime. Since W is a minimal spectral factor, then

\overline{W} _{+}^{−1}W_{−} = (\overline{W} _{+}^{−1}W)(W^{−1}W_{−}) = (\overline{Q}_{2}^{∗}K)Q_{1}^{∗}

is a minimal factorization, which implies the right coprimeness of K and Q_{1}. Since

\overline{W} _{+}^{−1}W_{−} = (\overline{W} _{+}^{−1}\overline{W})(\overline{W}^{−1}W_{−}) = \overline{Q}_{2}^{∗}(KQ_{1}^{∗})

is also a minimal factorization it follows that also \overline{W} is a minimal spectral factor. Hence, by Theorem 3.2, it follows that \overline{Q}_{2} is a right factor of Q_{−}.

That a LP pair determines factorizations of Q_{−} and Q_{+} is clear from the definition of LP pairs.

2. The proof is similar and hence omitted.

There are in general many minimal spectral factors associated with a given LP pair. In fact they are in a bijective correspondence with the set of inner factorizations of the inner function K. However two factorizations say of Q_{−} and K_{−} or of Q_{+} and K_{+} So a LP pair is insufficient to determine a unique, minimal spectral factor. However a primal and dual LP pairs taken together determine a unique, minimal spectral factor. This is the content of the next theorem.

Theorem 3.4 A minimal spectral factor determines uniquely its primal and dual Lindquist-Picci pairs. Conversely, a primal and dual Lindquist-Picci pair determine a unique, minimal spectral factor.

Proof: That a minimal spectral factor determines uniquely its Lindquist-Picci pair was shown in the proof of Proposition 3.2. The dual Lindquist-Picci pair is determined in the same way via duality considerations.

For this part of the proof it may be convenient to refer to the following diagram.

[figure: ^{\overline{W}^{−}}; ^{\overline{W}^{+}}; W_{+}; W_{−}; W; W_{1}; \overline{W}; W_{δ}; ^{\overline{W}^{δ}}; U; V; ^{\overline{K}^{1}}; ^{\overline{K}^{2}}; K_{1}; K_{2}; ^{\overline{Q}^{2}}; ^{\overline{Q}^{1}}; Q_{1}; Q_{2}; \overline{U}; \overline{V}]

Assume we are given a Lindquist-Picci pair (W, \overline{W}) and a dual Lindquist-Picci pair (W_{δ}, \overline{W}_{δ}). Since W is a minimal, stable spectral factor, there exists, by Theorem 3.2, a factorization Q_{+} = Q_{1}Q_{2} for which W = W_{−}Q_{1}. Similarly, since W_{δ} is a minimal, minimum phase spectral factor, there exists a factorization K = K_{1}K_{2} for which W_{δ}K_{2} = W_{−}. Now we determine normalized inner functions U and \overline{U} via the DSS factorization \overline{Q}_{1}^{∗}K_{1} = U\overline{U}^{∗}. This implies the equality

K_{1}\overline{U} = \overline{Q}_{1}U.  (28)

We define now

W_{1} = W_{δ}\overline{U}.  (29)

Then

\begin{aligned}W_{1} & = & W_{δ}\overline{U} = \overline{W}_{−}K_{1}\overline{U} \\ & = & \overline{W}_{−}\overline{Q}_{1}U = \overline{W} U \end{aligned}

Obviously W_{1} is a spectral factor. To show its minimality it suffices to exhibit a LP pair for it. In fact we will show now that (W, \overline{W}) is a LP pair for W_{1} and that (W_{δ}, \overline{W}_{δ}) is a dual LP pair for W_{1}. to do this it suffices to show the existence of normalized inner functions V and \overline{V} for which the previous diagram is commutative.

Since (W, \overline{W}) is a LP pair, K = \overline{W}^{−1}W is an inner function. Now

K = \overline{W}^{−1}W = (\overline{W}^{−1}\overline{W}_{−})(\overline{W} _{−}^{−1}W_{−})(W_{−}^{−1}W)
= \overline{Q}_{1}^{−1}K_{−}Q_{1} = \overline{Q}_{1}^{−1}K_{1}K_{2}Q_{1}
= U\overline{U}^{−1}K_{2}Q_{1}.

Now, by construction, U and \overline{U} are right coprime. So, using Proposition 2.1, we conclude that there exists an inner function V for which

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

Similarly, as (W_{δ}, \overline{W}_{δ}) is a dual LP pair, Q = W_{δ}^{−1}\overline{W}_{δ} is an inner function. Now

Q = W_{δ}^{−1}\overline{W}_{δ} = (W_{δ}^{−1}\overline{W}_{−})(\overline{W} _{−}^{−1}\overline{W}_{+})(\overline{W} _{+}^{−1}\overline{W}_{δ})
= K_{1}^{−1}Q_{−}\overline{K}_{1} = K_{1}^{−1}\overline{Q}_{1}\overline{Q}_{2}\overline{K}_{1}
= \overline{U}U^{−1}\overline{Q}_{2}\overline{K}_{1}

Hence, again by Proposition 2.1, there exists an inner function \overline{V} for which

\overline{Q}_{2}\overline{K}_{1} = U\overline{V} .  (31)

or Q = \overline{U}\overline{V}. Now

\overline{W}^{−1}W_{+} = (\overline{W}^{−1}\overline{W}_{+})(\overline{W} _{+}^{−1}\overline{W}_{δ})(\overline{W} _{δ}^{−1}W_{+}) = \overline{Q}_{2}\overline{K}_{1}\overline{K}_{2}.

On the other hand

\overline{W}^{−1}W_{+} = (\overline{W}^{−1}W_{1})(W_{1}^{−1}W)(W^{−1}W_{+}) = UV Q_{2}.

So we get the equality \overline{Q}_{2}\overline{K}_{1}\overline{K}_{2} = UV Q_{2}. Using the equality \overline{Q}_{2}\overline{K}_{1} = U\overline{V} we obtain

\overline{V} \overline{K}_{2} = V Q_{2}.  (32)

Equations (28), (30), (31) and (32), taken together, prove that the diagram is indeed a commutative diagram.

It is clear from the preceeding theorem, since LP pairs are in correspondence with factorizations of Q_{+} (or Q_{−}) and dual LP pairs in correspondence with factorizations of K_{−} (or K_{+}), that it is of interest to develop a comprehensive theory of factorizations of inner functions. This is taken up in the next section.

4 Factorization of inner functions

In this section we will use the standard, state space characterization of all-pass, and in particular inner functions, in order to derive a complete description of all left and right inner factors. This parametrization is based on the lattice of invariant subspaces of certain operators, or alternatively, on the set of symmetric solutions of certain Riccati equations. The connection between factorizations of inner functions and invariant subspaces of model operators is well known in Operator Theory. It originates in the work of Livsic on characteristic functions. In this connection see Debranges [1970 Brodskii and Livsic [1958] Helson [1964], Sz.-Nagy and Foias [1970] to cite a few. The correspondence between invariant subspaces and the factorization of polynomial matrices is due to A.C. Antoulas [1979] and Fuhrmann and Willems [1980]. In the context of System Theory the connection between invariant subspaces and solvability of the algebraic Riccati equation probably is due to Willems [1971]. The analysis of factors of inner functions in terms of invariant subspaces is due to Finesso and Picci [1982] whereas the connection to Riccati equations is made in Picci and Pinzoni [1993].

It is clear, in the scalar case, that a pole structure, given in terms of a stable polynomial, determines a unique inner functions of the same McMillan degree, providing we don’t distinguish between inner functions that differ by a factor of modulus 1. It is convenient to normalize the inner function by requiring that its value at infinity is equal to I. There are other inner functions derived in the same way but of lower McMillan degree if we allow pole-zero cancellations. This result extends in a natural way to the multivariable case. This is the content of the next theorem. We will outline two different approaches to the problem, a state space approach and a polynomial approach.

From the state space point of view, utilizing realization theory, the left pole structure

of a given rational function G, with minimal realization ( \begin{matrix}A & B \\ C & D \end{matrix} ), is determined via the

observable pair (A, C). From this point of view the problem is to find extensions of the pair (A, C) to a realization, not necessarily minimal, of an inner function. Since an inner function in H_{+}^{∞} is stable, such a generator A is necessarily stable.

In a similar way, a stable polynomial matrix D determines a set of inner functions, namely the inner function that have D as a left denominator matrix. That these two results are related is obvious since a nonsingular polynomial matrix D determines a unique, up to isomorphism, observable pair (A, C) and such an observable pair determines a nonsingular polynomial matrix. This is best seen through a polynomial coprime factorization of a state/output transfer function, namely

C(sI − A)^{−1} = D(s)^{−1}H(s).

Such a left denominator matrix D is uniquely determined up to a left unimodular factor.

Theorem 4.1 1. Given an observable pair (A, C), with A stable. Then ( \begin{matrix}A & B \\ C & I \end{matrix} ) is a

normalized inner function if and only if B = −Y C^{∗} with Y a nonnegative solution of the homogeneous Riccati equation

AY + Y A^{∗} + Y C^{∗}CY = 0.  (33)

The realization ( \begin{matrix}A & B \\ C & I \end{matrix} ) is observable. The reachable subspace is given by ImY. The

inner function of maximal McMillan degree so obtainable corresponds to the positive definite solution Y_{+} of (33).

2. Let D be a square, nonsingular, stable polynomial matrix. Then U = D^{−1}N is an inner function if and only if there exist factorizations D = D_{γ}D_{1} and N = D_{γ}N_{1} with N_{1} the antistable solution of the polynomial spectral factorization problem

N_{1}N_{1}^{∗} = D_{1}D_{1}^{∗}.  (34)

The inner function of maximal McMillan degree so obtainable corresponds to D_{γ} = I.

Proof:

1. Assume Y is a solution of the Riccati equation (33). Then U = ( \begin{matrix}A & −Y C^{∗} \\ C & I \end{matrix} ) and

U^{∗} = ( \begin{matrix}−A^{∗} & C^{∗} \\ CY & I \end{matrix} ). This implies

\begin{aligned}U^{∗}U & = & ( \begin{matrix}−A^{∗} & C^{∗} \\ CY & I \end{matrix} ) × ( \begin{matrix}A & −Y C^{∗} \\ C & I \end{matrix} ) = ( \begin{matrix}A & 0 & −Y C^{∗} \\ C^{∗}C & −A^{∗} & C^{∗} \\ C & CY & I \end{matrix} ) \\ & = & ( \begin{matrix}A + Y C^{∗}C & 0 & 0 \\ C^{∗}C & −A^{∗} − C^{∗}CY & C^{∗} \\ C & 0 & I \end{matrix} ) = I \end{aligned}

In the computation we used an isomorphism of the form ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) and the fact that

Y solves (33).

Conversely, suppose U = ( \begin{matrix}A & B \\ C & I \end{matrix} ) is inner. Let us consider a left coprime factoriza-

tion of the state to output transfer function

C(sI − A)^{−1} = D(s)^{−1}H(s).  (35)

The polynomial matrix H has the property that its columns are a basis for the polynomial model X_{D}, see Hautus and Heymann [1979], Wimmer [1979] or Fuhrmann [1983]. Thus it follows that N(s) = D(s) + H(s)B. Since U is inner we have

\begin{aligned}0 & = & NN^{∗} − DD^{∗} = (D + HB)(D^{∗} + B^{∗}H^{∗}) − DD^{∗} \\ & = & H(s)BD^{∗} + DB^{∗}H(s)^{∗} + H(s)BB^{∗}H(s)^{∗} \end{aligned}

This implies the equality

C(sI − A)^{−1}B − B^{∗}(sI + A^{∗})^{−1}C^{∗} − C(sI − A)^{−1}BB^{∗}(sI + A^{∗})^{−1}C^{∗} = 0 (36)

Equating coefficients of expansion at infinity, we have CB = B^{∗}C^{∗}. Moreover we can conclude, by observability, that

ImB^{∗} ⊂ ImC  (37)

Thus there exists a constant map Y_{1} for which B^{∗} = CY_{1}. We prefer to write this as

B = −Y C^{∗}.  (38)

Upon substituting this in (36) we get

\begin{aligned}0 & = & C(sI − A)^{−1}Y C^{∗} − CY^{∗}(sI + A^{∗})^{−1}C^{∗} + C(sI − A)^{−1}Y C^{∗}CY^{∗}(sI + A^{∗})^{−1}C^{∗} \\ & = & C(sI − A)^{−1}[−(sI − A)Y^{∗} + Y (sI + A^{∗}) + Y C^{∗}CY^{∗}](sI + A^{∗})^{−1}C^{∗} \end{aligned}

and hence, by the observability of (A, C), that

−(sI − A)Y^{∗} + Y (sI + A^{∗}) + Y C^{∗}CY^{∗} = 0

Equating coefficients, we have

\begin{aligned}Y^{∗} & = & Y \\ AY + Y A^{∗} + Y C^{∗}CY & = & 0 \end{aligned}

Since U^{−1} is antistable, it follows that A + Y C^{∗}C is antistable. This implies that Y > 0.

To compute the reachable subspace of this realization, we rewrite the Riccati equation (33) as

AY = −Y (A^{∗} + CC^{∗}Y).

This implies

∑_{i} A^{i}Bξ_{i} = − ∑_{i} A^{i}Y C^{∗}ξ_{i} = −Y ∑_{i}(−1)^{i}(A^{∗} + CC^{∗}Y)^{i}C^{∗}ξ_{i}

which by the observability of the pair (A+Y C^{∗}C, C), shows that the reachable subspace is indeed ImY.

2. Assume U = D^{−1}N is inner. Let D_{γ} be the greatest common left divisor of D and N. Then the factorizations D = D_{γ}D_{1} and N = D_{γ}N_{1} exist and moreover we have U = D^{−1}N = D_{1}^{−1}N_{1}. As U is inner, we have N_{1}N_{1}^{∗} = D_{1}D_{1}^{∗}.

Conversely, assume D is stable and nonsingular. Let D = D_{γ}D_{1} be a factorization into nonsingular factors. Let N_{1} be the antistable solution of the polynomial spectral factorization problem

N_{1}N_{1}^{∗} = D_{1}D_{1}^{∗}.  (39)

normalized so that D_{1}^{−1}N_{1}(∞) = I. Then U = D_{1}^{−1}N_{1} is inner and hence can also be written as U = D^{−1}N where N = D_{γ}N_{1}.

Of course a dual result can be stated in terms of reachable pairs or of right denominator polynomial matrices.

As a corollary to the previous theorem we can state the following result, giving a well known characterization of rational inner functions. See Finesso and Picci [1982] and Glover [1984], and also Fuhrmann and Ober [1993] where a slightly more general result is derived. Of course one can give a direct proof of this result based on the state space isomorphism theorem. This is the approach taken in the previously mentioned papers.

Theorem 4.2 1. A function U is rational and inner in H_{+}^{∞} if and only if it has a minimal state space realization of the form

U = ( \begin{matrix}A & B \\ −B^{∗}X_{+} & I \end{matrix} )  (40)

with A stable and X_{+} the unique, positive definite solution of the homogeneous ARE

A^{∗}X + XA + XBB^{∗}X = 0  (41)

2. A function U is rational and inner in H_{+}^{∞} if and only if it has a minimal state space realization of the form

U = ( \begin{matrix}A & −Y_{+}C^{∗} \\ C & I \end{matrix} )  (42)

with A stable and Y_{+} the unique, positive definite solution of the homogeneous ARE

AY + Y A^{∗} + Y C^{∗}CY = 0  (43)

Of course a natural question to ask is for the characterization and parametrization of all left and right inner factors of the inner function U. We discuss briefly the history of ideas related to this. It has been known for a long time, in fact it is one of the reasons for the study of characteristic functions in general and inner functions in particular, that there is a very close connection between factorization of inner functions and the invariant subspaces of a corresponding model operator. This is easier to describe in the case of the disc, i.e. for discrete time operators. Thus, if U is a square matrix inner function in the disc, then H(U) = {UH^{2}}^{⊥} is a backward shift invariant subspace of H^{2}. Defining an operator S_{U} : {UH^{2}}^{⊥} ⟶ {UH^{2}}^{⊥} by

S_{U}f = P_{{UH^{2}}^{⊥}}zf  f ∈ {UH^{2}}^{⊥}  (44)

then there is a bijective correspondence between invariant subspaces of S_{U} and factorizations U = U_{1}U_{2} into the product of square inner functions. This, provided we identify two left inner factors of U that differ only by a constant left unitary factor. The correspondence is given by assigning to the factorization U = U_{1}U_{2} the S_{U}-invariant subspace V = U_{1}{U_{2}H^{2}}^{⊥}.

The situation is somewhat more complex in the continuous time case. Here the underlying space is H_{+}^{2} and in this space the ”multiplication by s” is an unbounded operator. There are two ways to bypass this difficulty. Either we redefine our notion of invariant subspace to be one that is invariant under multiplication by all, scalar, H_{+}^{∞} functions. Or, alternatively, to use rationality in order to legitimize the ”multiplication by s” operator.

We proceed to elaborate on this. We begin by recalling the basic elements of the shift realization, see Fuhrmann [1976,1977, 1983] for more details.

Suppose we are given a p×m function g defined in [0, ∞), for which its Laplace transform G is rational in H_{+}^{2}. This means of course that the entries of g are exponential polynomials with all exponents having negative real part. A minimal, time domain, realization of g is obtained by taking the state space M to be the smallest left translation invariant subspace of L^{2}(0, ∞) that contains all columns of g. The realization is defined by

\{ \begin{matrix}(e^{At}f)(x) & = & f(x + t) \\ Bξ & = & g(t)ξ \\ Cf & = & f(0) \end{matrix}  (45)

Note that the infinitesimal generator A of the left translation semigroup is given by Af = f′. Restricted to our, finite dimensional, space M, this operator is actually bounded. Thus the realization can be written as

\{ \begin{matrix}Af & = & f′ \\ Bξ & = & g(t)ξ \\ Cf & = & f(0) \end{matrix}  (46)

Note, that by Laplace transform theory, we have, with Lf = F, that

Lf′ = sF − f(0)

Also we have

f(0) = lim_{s→∞} sF (s)

To see this note that

sF (s) = s ∫_{0}^{∞} e^{−st}f(t)dt
= −s [\frac{1}{s}e^{−st}f(t)]_{0}^{∞} + s ∫_{0}^{∞} \frac{1}{s}e^{−st}f′(t)dt
= f(0) − ∫_{0}^{∞} e^{−st}f′(t)dt

where we have used the fact that the Laplace transform of a trigonometric polynomial is a strictly proper rational functions and hence its value at infinity is zero.

Thus, in the frequency domain, the shift realization can be written as

\{ \begin{matrix}Af(s) & = & sf(s) − lim_{s→∞} sf(s) \\ Bξ & = & G(s)ξ \\ Cf & = & lim_{s→∞} sf(s) \end{matrix}  (47)

Now assume that G = ND^{−1} is a polynomial right coprime factorization. Then the state space M of a minimal realization of G can be taken to be the smallest subspace of H_{+}^{2} which is invariant under the map f ↦ sf(s) − lim_{s→∞} sf(s). Since the elements of M are in our case strictly proper rational functions, this map is equivalent to f ↦ S_{−}f = π_{−}sf. For the full explanation of these maps we refer to the above mentioned papers by the author.

Next we use the fact that a finite dimensional subspace M of H_{+}^{2} which satisfies the invariance condition S_{−}M ⊂ M is necessarily of the form M = X^{D} for some nonsingular polynomial matrix D. The inclusion M ⊂ H_{+}^{2} implies that D is necessarily stable. Thus the realization (47) is the same as the algebraic shift realization given in Fuhrmann [1976b,1983].

Note also that the reachability map of the realization (47) is given by the Hankel operator Ĥ_{G} : H_{−}^{2} ⟶ H_{+}^{2} defined by Ĥ_{G}h = P_{+}Gh where P_{+} is the orthogonal projection of L^{2} onto H_{+}^{2}. Now M = ImĤ_{G} and hence M^{⊥} = KerH_{G}^{∗} = KerH_{G^{∗}}. Now if U is a rational inner functions, normalized by U(∞) = I, then U = I + G with G strictly proper. Clearly H_{G} = H_{U} and hence also

KerH_{G^{∗}} = KerH_{U^{∗}} = UH_{+}^{2}

So to conclude we have

M = {UH_{+}^{2}}^{⊥} = X^{D}  (48)

The inner function U determines, and in return is also determined, by the polynomial matrix D. In fact it is easy to check that U = D^{−1}\overline{D} where \overline{D} is the antistable solution to the spectral factorization problem \overline{D}\overline{D}^{∗} = DD^{∗}.

Suppose now that the inner function has a factorization of the form U = U_{1}U_{2}, with U_{i} inner and U_{i}(∞) = I. This of course implies the inclusions U_{1}U_{2}H_{+}^{2} ⊂ U_{1}H_{+}^{2} and, equivalently, {U_{1}H_{+}^{2}}^{⊥} ⊂ {U_{1}U_{2}H_{+}^{2}}^{⊥}. This can be seen also from a different perspective. Let U_{1} = N_{1}\overline{D}_{1}^{−1} and U_{2} = \overline{N}_{2}D_{2}^{−1} and let N_{2}D_{1}^{−1} = \overline{D}_{1}^{−1}\overline{N}_{2}. So

\begin{aligned}U & = & N_{1}\overline{D}_{1}^{−1}\overline{N}_{2}D_{2}^{−1} = N_{1}N_{2}D_{1}^{−1}D_{2}^{−1} \\ & = & (N_{1}N_{2})(D_{2}D_{1})^{−1} \end{aligned}

Using equation (48), we have the inclusion

X^{D_{2}D_{1}} ⊃ X^{D_{1}}

This of course is consistent with the representation of submodules of rational models, see Fuhrmann and Willems [1980].

There exists however another interesting connection, namely the one between invariant subspaces and the Riccati equation. This goes back to the classic paper Willems [1971]. In this paper Willems parametrizes the set of solutions of an ARE via the set of invariant subspaces of a certain related operator. For more on this theme, as well as connections to factorization theory, we refer to Finesso and Picci [1982] and Fuhrmann [1985,1989].

In view of the preceeding remarks, it is natural to develop a factorization theory for inner functions that would unify arithmetic (i.e. factorization), geometric (i.e. invariant subspaces) and dynamic (i.e. Riccati equations oriented) approaches. This leads to the following.

Theorem 4.3 1. Let U be a square inner function in the right half plane, normalized so that U(∞) = I. Let

U = ( \begin{matrix}A & −Y_{+}C^{∗} \\ C & I \end{matrix} )  (49)

be a minimal realization with A stable and Y_{+} the unique, positive definite solution of the homogeneous ARE

AY + Y A^{∗} + Y C^{∗}CY = 0  (50)

Then there exists a bijective correspondence between

(a) Normalized left inner factors of U.

(b) Invariant subspaces of A.

(c) Nonnegative definite solution of the homogeneous ARE (50).

The correspondence is given by

U_{1} = ( \begin{matrix}A & −Y C^{∗} \\ C & I \end{matrix} )  (51)

where Y is a nonnegative definite solution of the homogeneous ARE (50), and V = ImY ⊂ R^{n} is A invariant, in fact V is the reachable subspace of the realization (51).

2. Let U be a square inner function in the right half plane, normalized so that U(∞) = I. Let

U = ( \begin{matrix}A & B \\ −B^{∗}X & I \end{matrix} )  (52)

be a minimal realization with A stable and X the unique, positive definite solution of the homogeneous ARE

A^{∗}X + XA + XBB^{∗}X = 0.  (53)

Then there exists a bijective correspondence between

(a) Normalized right inner factors of U.

(b) Nonnegative definite solution of the homogeneous ARE (53).

(c) Invariant subspaces of A.

The correspondence is given by

U_{2} = ( \begin{matrix}A & B \\ −B^{∗}X_{2} & I \end{matrix} )  (54)

where X_{2} is a nonnegative definite solution of the homogeneous ARE (53). and V = KerX ⊂ R^{n} is A invariant, in fact it is the unobservable subspace of the realization (54).

Proof:

1. Because of the interest in the connection between the state space and the functional approaches we give multiple proofs of several implications.

(1a) ⇒ (1b) Assume there exists a factorization U = U_{1}U_{2}, with U_{i} normalized inner functions. If U_{1} = D_{1}^{−1}\overline{N}_{1} and U_{2} = \overline{D}_{2}^{−1}N_{2} are polynomial coprime factorizations and D_{2}^{−1}N_{1} is a polynomial left coprime factorization of \overline{N}_{1}\overline{D}_{2}^{−1}, then

\begin{aligned}U & = & U_{1}U_{2} = D_{1}^{−1}\overline{N}_{1}\overline{D}_{2}^{−1}N_{2} \\ & = & D_{1}^{−1}D_{2}^{−1}N_{1}N_{2} = (D_{2}D_{1})^{−1}(N_{1}N_{2}) \end{aligned}

We note that the factorization (D_{2}D_{1})^{−1}(N_{1}N_{2}) is necessarily left coprime. The state space of the shift realization of U is X^{D_{2}D_{1}} and we obviously have the inclusion X^{D_{1}} ⊂ X^{D_{2}D_{1}}. Moreover X^{D_{1}} is S^{D_{2}D_{1}} invariant. Since, by the state space isomorphism theorem, the shift realization and the realization (49) are isomorphic, then X^{D_{1}} determines a unique A-invariant subspace.

We give also a state space proof of this implication. Let U = U_{1}U_{2} be factorization with

U_{i} normalized inner functions. Assume U_{i} = ( \begin{matrix}A_{i} & −Y_{i}C_{i}^{∗} \\ C_{i} & I \end{matrix} ) be minimal realizations

in the state space X_{i}, with Y_{i} the positive definite solution of the Riccati equation A_{i}Y_{i} + Y_{i}A_{i}^{∗} + Y_{i}C_{i}^{∗}C_{i}Y_{i} = 0. By series composition we have

\begin{aligned}U & = & ( \begin{matrix}A_{1} & −Y_{1}C_{1}^{∗} \\ C_{1} & I \end{matrix} ) × ( \begin{matrix}A_{2} & −Y_{2}C_{2}^{∗} \\ C_{2} & I \end{matrix} ) \\ & = & ( \begin{matrix}A_{2} & 0 & −Y_{2}C_{2}^{∗} \\ −Y_{1}C_{1}^{∗}C_{2} & A_{1} & −Y_{1}C_{1}^{∗} \\ C_{2} & C_{1} & I \end{matrix} ) = ( \begin{matrix}A & −Y_{+}C^{∗} \\ C & I \end{matrix} ) \end{aligned}  (55)

Note that, using the results of Callier and Nahum [1975], this realization is minimal as there is no possibility of zero-pole cancellations. This clearly shows that V =

\{( \begin{matrix}0 \\ x \end{matrix} ) |x ∈ X_{1}\} is an ( \begin{matrix}A_{2} & 0 \\ −Y_{1}C_{1}^{∗}C_{2} & A_{1} \end{matrix} ) invariant subspace of X_{2} ⊕ X_{1}.

(1b) ⇒ (1a) Next we show that an A-invariant subspace V determines uniquely a factorization of U. Assume U = D^{−1}N is a left coprime factorization. The associated state space of this realization is the rational model space X^{D}. By the state space isomorphism theorem, there exists an S^{D} invariant subspace of X^{D}. In fact this subspace can be easily described precisely. We have C(sI − A)^{−1} = D(s)^{−1}H(s). This H has the property that its columns are a basis for the polynomial model X_{D}, see Wimmer [1979], Hautus and Heymann [1979] or Fuhrmann [1983], and hence the columns of D(s)^{−1}H(s) are a basis of X^{D}. Thus the S^{D} invariant subspace is given by {D(s)^{−1}H(s)v|v ∈ V}. Now any S^{D} invariant subspace of X^{D} is of the form X^{D_{1}} where D_{1} is a right nonsingular factor of D, i.e. we can write D = D_{2}D_{1}. The orthogonal complement of X^{D_{1}} is an invariant subspace and hence has a representation of the form (X^{D_{1}})^{⊥} = U_{1}H_{+}^{2} for some inner function U_{1} which without loss of generality we can assume to be normalized. The inclusion X^{D_{1}} ⊂ X^{D_{2}D_{1}} implies (X^{D_{2}D_{1}})^{⊥} ⊂ (X^{D_{1}})^{⊥}, i.e. UH_{+}^{2} ⊂ U_{1}H_{+}^{2}. This last inclusion implies the existence of a, necessarily normalized inner function U_{2} for which the factorization U = U_{1}U_{2} holds.

(1b) ⇒ (1c) Let V be an A-invariant subspace. If V is a basis matrix for V, then there exists a unique matrix A_{1} for which

AV = V A_{1}.  (56)

We set C_{1} = CV. The pair (A_{1}, C_{1}) is observable as C_{1}A_{1}^{n} = CA^{n}V, which implies that ∩_{i=0}^{∞}C_{1}A_{1}^{i} = ∩_{i=0}^{∞}CA^{i}V = kerV = 0, using the fact that V is injective. Let Y_{1} be the unique positive definite solution of the Riccati equation

A_{1}Y + Y A_{1}^{∗} + Y C_{1}^{∗}C_{1}Y = 0.

We set now Y = V Y_{1}V^{∗}. We check now that Y is a solution of the Riccati equation (33).

AY + Y A^{∗} + Y C^{∗}CY = AV Y_{1}V^{∗} + V Y_{1}V^{∗}A^{∗} + V Y_{1}V^{∗}C^{∗}CV Y_{1}V^{∗}
= V A_{1}Y_{1}V^{∗} + V Y_{1}A_{1}^{∗}V^{∗} + V Y_{1}C_{1}^{∗}C_{1}Y_{1}V^{∗}
= V [A_{1}Y_{1} + Y_{1}A_{1}^{∗} + Y_{1}C_{1}^{∗}C_{1}Y_{1}]V^{∗} = 0

(1c) ⇒ (1b) Assume now that Y is a solution of the homogeneous Riccati equation (50). Rewriting the equation as

AY = −Y (A^{∗} + CC^{∗}Y)

it is clear that V = ImY is an A-invariant subspace of R^{n}.

Since from the previous equality we get A^{i}Y = (−1)^{i}Y (A^{∗} + C^{∗}CY)^{i}, it follows that

∑_{i} A^{i}Bξ_{i} = − ∑_{i} A^{i}Y C^{∗}ξ_{i}
= − ∑_{i}(−1)^{i}Y (A^{∗} + C^{∗}CY)^{i}C^{∗}ξ_{i}
= ImY

(1a) ⇒ (1c) We show now how a factorization leads to a solution of the Riccati equation

(50). Using the realization (55), it is easy to check that that Y = ( \begin{matrix}0 & 0 \\ 0 & Y_{1} \end{matrix} ) is a

solution of the Riccati equation (50). To see this we compute

( \begin{matrix}A_{2} & 0 \\ −Y_{1}C_{1}^{∗}C_{2} & A_{1} \end{matrix} ) ( \begin{matrix}0 & 0 \\ 0 & Y_{1} \end{matrix} ) + ( \begin{matrix}0 & 0 \\ 0 & Y_{1} \end{matrix} ) ( \begin{matrix}A_{2}^{∗} & −C_{2}^{∗}C_{1}Y_{1} \\ 0 & A_{1}^{∗} \end{matrix} )

+ ( \begin{matrix}0 & 0 \\ 0 & Y_{1} \end{matrix} ) ( \begin{matrix}C_{2}^{∗}C_{2} & C_{2}^{∗}C_{1} \\ C_{1}^{∗}C_{2} & C_{1}^{∗}C_{1} \end{matrix} ) ( \begin{matrix}0 & 0 \\ 0 & Y_{1} \end{matrix} )

= ( \begin{matrix}0 & 0 \\ 0 & A_{1}Y_{1} + Y_{1}A_{1}^{∗} + Y_{1}C_{1}^{∗}C_{1}Y_{1} \end{matrix} ) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )

2. This results follows out of duality cosiderations. Note that the unobservable subspace of the realization (54) is given by

∩_{i}Ker(−B^{∗}X)A^{i} = ∩_{i}KerB^{∗}(A^{∗} + XBB^{∗})^{i}X = ∩_{i}KerB^{∗}(A^{∗})^{i}X = KerX.

There is a natural partial ordering of left inner factors of a given inner function. There is another natural partial order in the set of symmetric solutions of the Riccati equation (33). It is not surprising that these partial orders are related through the following theorem. The proof is based on previous results and hence omitted.

Theorem 4.4 1. Let U(s) be inner, and let U(s) = I − C(sI − A)^{−1}Y_{+}C^{∗} with Y_{+} > 0 the unique, positive definite solution of the ARE

AY + Y A^{∗} + Y C^{∗}CY = 0.  (57)

Let U_{i}(s) = I −C_{i}(sI −A)^{−1}Y_{i}C_{i}^{∗}, i = 1, 2 be two left factors of U with Y_{i} nonnegative definite solutions of the Riccati equation (57). Then U_{2} is a left inner factor of U_{1} if and only if Y_{2} ≤ Y_{1}.

2. Let U(s) be inner, and let U(s) = I −B^{∗}X(sI −A)^{−1}B with X > 0 the unique, positive definite solution of the ARE

A^{∗}X + XA + XBB^{∗}X = 0.  (58)

Let U_{i}(s) = I−B_{i}^{∗}X_{i}(sI−A)^{−1}B_{i}, i = 1, 2 be two left factors of U with X_{i} nonnegative definite solutions of the Riccati equation (58). Then U_{2} is a right inner factor of U_{1} if and only if X_{2} ≤ X_{1}.

As a corollary to the previous results we get.

Corollary 4.1 1. Let U, U_{1} be rational inner functions with the polynomial left coprime factorizations

\{ \begin{matrix}U & = & D^{−1}E \\ U_{1} & = & D_{1}^{−1}E_{1} \end{matrix}

then U_{1} is a left inner factor of U if and only if D_{1} is a right factor of D.

2. Let U, U_{1} be rational inner functions with the polynomial right coprime factorizations

\{ \begin{matrix}U & = & ED^{−1} \\ U_{1} & = & E_{1}D_{1}^{−1} \end{matrix}

then U_{1} is a right inner factor of U if and only if D_{1} is a left factor of D.

5 State space computation of the DSS factorization

In the sequel we will need to use the existence of stabilizing solutions of the algebraic Riccati equation

A^{∗}X + XA − XBB^{∗}X = 0  (59)

in the general case when A may have its eigenvalues in both the open left and right half planes. We recall, from Fuhrmann and Ober [1993], that one method that could be used to prove the existence of stabilizing solutions to a variety of algebraic Riccati equations is via the computation of, appropriately normalized, coprime factorizations over H_{+}^{∞}. In particular, for the class of rational, antistable transfer functions this lead to the DSS factorization. Now, as we saw in section 2, the DSS factorization exists not only for rational antistable functions but for the larger class of strictly noncyclic functions. These are L^{∞} functions for which the kernel and image of the induced Hankel operator are associated with square inner functions. A very special case, but the case of interest for us, is that of rational functions bounded on the imaginary axis, i.e. having no poles there. For this class of functions we can compute specific state space formulas for the DSS factorization.

Theorem 5.1 Let

G = ( \begin{matrix}A & B \\ C & D \end{matrix} )  (60)

be a proper, rational function in L^{∞}. Let

G = NM^{−1} = \overline{M}^{−1}\overline{N}  (61)

be the normalized DSS factorizations of G. Then

1. The right factors have the state space realization

( \begin{matrix}M \\ N \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \\ C − DB^{∗}X & D \end{matrix} )  (62)

where X is the unique, stabilizing solution, i.e. A − BB^{∗}X is stable, of the algebraic Riccati equation

A^{∗}X + XA = XBB^{∗}X.  (63)

The McMillan degree of M, δ(M), is given by

δ(M) = n − dim ker X.  (64)

2. The left factors have the state space realization

( \overline{N} \overline{M} ) = ( \begin{matrix}A − ZC^{∗}C & B − ZC^{∗}D & −ZC^{∗} \\ C & D & I \end{matrix} )  (65)

where Z is the unique, stabilizing solution, i.e. A − ZC^{∗}C is stable, of the algebraic Riccati equation

ZA^{∗} + AZ = ZC^{∗}CZ.  (66)

The McMillan degree of \overline{M}, δ(\overline{M}), is given by

δ(\overline{M}) = dim ImZ.  (67)

Proof:

1. Let G = ED_{1}^{−1} be a polynomial right coprime factorization. Let D_{1} = D_{+}D_{−} be a factorization of D into an antistable factor D_{+} and a stable factor D_{−}. Note that, as G ∈ L^{∞}, there are no poles on the imaginary axis. Thus we have

G = ED_{1}^{−1} = E(D_{+}D_{−})^{−1} = ED_{−}^{−1}D_{+}^{−1}  (68)

It is easy to check that KerH_{G} = D_{+}\overline{D}_{−}^{−1}, where \overline{D}_{−} is the stable solution to the polynomial spectral factorization problem

D_{+}^{∗}D_{+} = \overline{D}_{−}^{∗}\overline{D}_{−}  (69)

normalized so that D_{+}\overline{D}_{−}^{−1}(∞) = I. Therefore we can write

G = ED_{1}^{−1} = ED_{−}^{−1}D_{+}^{−1} = [E(\overline{D}_{−}D_{−})^{−1}][D_{+}\overline{D}_{−}^{−1}]^{−1}.  (70)

Clearly, as \overline{D}_{−}D_{−} is stable, we have E(\overline{D}_{−}D_{−})^{−1} ∈ H_{+}^{∞}, whereas D_{+}\overline{D}_{−}^{−1} is inner by construction. At the price of losing coprimeness we can write

M = D_{+}\overline{D}_{−}^{−1} = D_{+}D_{−}(\overline{D}_{−}D_{−})^{−1}  (71)

and hence we can compare

\{ \begin{matrix}M^{−1} & = & (\overline{D}_{−}D_{−})(D_{+}D_{−})^{−1} \\ G & = & E(D_{+}D_{−})^{−1} \end{matrix}  (72)

Since the last factorization is coprime and the realization (60) is assumed minimal, we can conclude that M^{−1} has a realization

M^{−1} = ( \begin{matrix}A & B \\ C_{0} & I \end{matrix} )

and hence

M = ( \begin{matrix}A − BC_{0} & B \\ −C_{0} & I \end{matrix} ) = ( \begin{matrix}A_{1} & B \\ −C_{0} & I \end{matrix} )

Since obviously the pair (A_{1}, B) is reachable and moreover A_{1} is stable, then there exists, by Theorem 4.2, a unique positive definite solution X_{+} of the algebraic Riccati equation A_{1}^{∗}X + XA_{1} + XBB^{∗}X = 0. This corresponds to the maximal inner function reconstructible from (A_{1}, B) via Theorem 4.1.

Now, obviously, M is a right factor of M_{1} = ( \begin{matrix}A_{1} & B \\ −B^{∗}X_{+} & I \end{matrix} ), where X_{+} is the positive

definite solution of the ARE

A_{1}^{∗}X + XA_{1} = −XBB^{∗}X.  (73)

Therefore M has, by Theorem 4.3, a representation of the form

M = ( \begin{matrix}A_{1} & B \\ −B^{∗}X & I \end{matrix} )

where X is a nonnegative definite solution of the ARE (73). Comparing the two realizations

M = ( \begin{matrix}A_{1} & B \\ −B^{∗}X & I \end{matrix} ) = ( \begin{matrix}A_{1} & B \\ −C_{0} & I \end{matrix} )

and using the reachability of the pair (A, B), we conclude that C_{0} = B^{∗}X. Substituting in the ARE (73), we get

(A^{∗} − XBB^{∗})X + X(A − BB^{∗}X) + XBB^{∗}X = 0

or

XA + A^{∗}X = XBB^{∗}X.

We can use Theorem 4.3 to conclude that the unobservable space of the realization (62) is KerX. Hence the McMillan degree of M is n − dim KerX.

Now

N = GM = ( \begin{matrix}A & B \\ C & D \end{matrix} ) × ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & 0 & B \\ −BB^{∗}X & A & B \\ C & −DB^{∗}X & D \end{matrix} )

Applying the isomorphism ( \begin{matrix}I & 0 \\ −I & I \end{matrix} ), we get

N = ( \begin{matrix}A − BB^{∗}X & 0 & B \\ 0 & A & 0 \\ C − DB^{∗}X & −DB^{∗}X & D \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & B \\ C − DB^{∗}X & D \end{matrix} )

The realizations of M and N can be combined to yield (62).

2. The proof is similar and is omitted.

6 State space computation of spectral factors

Our aim in this section is to derive state space formulas for the extremeal, minimal, spectral factors. Our approach utilizes the polynomial model approach to realization theory, as developed in Fuhrmann [1976] in conjunction with the Kalman state space isomorphism theorem. Of particular use is the Brockett and Rahimi [1972] version of the state space isomorphism theorem applicable to the case of Hamiltonian systems. In this case the isomorphism is given by a symplectic map. The method advocated here is simple yet very powerful. For similar reasoning in a different context see Fuhrmann and Ober [1993].

Our starting point is a minimal realization of the minimum phase, stable spectral factor W_{−}. With this we characterize the set of all minimal, stable spectral factors. This we do by analyzing the zero flipping from the left half plane to the right half plane.

Theorem 6.1 Let Φ be a coercive spectral function, normalized by Φ(∞) = I, and let

W_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} )  (74)

be a minimal realization of the stable, minimum phase spectral factor of Φ. Then

1. W is a minimal, stable spectral factor of Φ if and only if

W = ( \begin{matrix}A & B − XC^{∗} \\ C & I \end{matrix} )  (75)

where X is a nonnegative definite solution of the homogeneous Riccati equation

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0.  (76)

2. A state space realization of the maximum phase, stable spectral factors is given by

W_{+} = ( \begin{matrix}A & B − X_{+}C^{∗} \\ C & I \end{matrix} )  (77)

where X_{+} is the positive definite solution of the homogeneous Riccati equation (76).

Proof:

1. Let W_{−} = \overline{D}_{−}^{−1}\overline{E}_{−} be a polynomial, left coprime factorization with \overline{D}_{−}, \overline{E}_{−} stable. Since we want the factor W to have the same poles as W_{−}, we assume that W = \overline{D}_{−}^{−1}E. Since Φ = W_{−}W_{−}^{∗} = W W^{∗} it follows that

\overline{E}_{−}\overline{E}_{−}^{∗} = \overline{E}\overline{E}^{∗}

that is E can be obtained from \overline{E}_{−} by a process of polynomial spectral factorization. Comparing now

\{ \begin{matrix}W_{−} & = & \overline{D}_{−}^{−1}\overline{E}_{−} \\ W & = & \overline{D}_{−}^{−1}E \end{matrix}

we may assume that

W = ( \begin{matrix}A & B_{0} \\ C & I \end{matrix} )

From the equality W W^{∗} = W_{−}W_{−}^{∗} we get, using cascade connections, two minimal realizations of Φ, namely

Φ = ( \begin{matrix}−A^{∗} & 0 & C^{∗} \\ −BB^{∗} & A & B \\ −B^{∗} & C & I \end{matrix} ) = ( \begin{matrix}−A^{∗} & 0 & C^{∗} \\ −B_{0}B_{0}^{∗} & A & B_{0} \\ −B_{0}^{∗} & C & I \end{matrix} )

Necessarily, the two realizations are isomorphic. Since the two previous realizations

are Hamiltonian, the isomorphism matrix ( \begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ) is necessarily symplectic. The

isomorphism equations are

\{ \begin{matrix}( \begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ) ( \begin{matrix}C^{∗} \\ B \end{matrix} ) = ( \begin{matrix}C^{∗} \\ B_{0} \end{matrix} ) \\ ( \begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ) ( \begin{matrix}−A^{∗} & 0 \\ −BB^{∗} & A \end{matrix} ) = ( \begin{matrix}−A^{∗} & 0 \\ −B_{0}B_{0}^{∗} & A \end{matrix} ) ( \begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ) \\ ( −B^{∗} C ) = ( −B_{0}^{∗} C ) ( \begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ) \end{matrix}

From the equation X_{12}A = −A^{∗}X_{12}, noting that σ(A) ∩ σ(A^{∗}) = ∅, it follows that X_{12} = 0. Using the observability of the pair (A, C), we conclude from the equalities −X_{11}A^{∗} − X_{12}BB^{∗} = −A^{∗}X_{11} and X_{11}C^{∗} = C^{∗}, that X_{11} = I.

Thus we are reduced to

\{ \begin{matrix}( \begin{matrix}I & 0 \\ X_{21} & X_{22} \end{matrix} ) ( \begin{matrix}C^{∗} \\ B \end{matrix} ) = ( \begin{matrix}C^{∗} \\ B_{0} \end{matrix} ) \\ ( \begin{matrix}I & 0 \\ X_{21} & X_{22} \end{matrix} ) ( \begin{matrix}−A^{∗} & 0 \\ −BB^{∗} & A \end{matrix} ) = ( \begin{matrix}−A^{∗} & 0 \\ −B_{0}B_{0}^{∗} & A \end{matrix} ) ( \begin{matrix}I & 0 \\ X_{21} & X_{22} \end{matrix} ) \\ ( −B^{∗} C ) = ( −B_{0}^{∗} C ) ( \begin{matrix}I & 0 \\ X_{21} & X_{22} \end{matrix} ) \end{matrix}

In particular we have the pair of equalities

\{ \begin{matrix}CX_{22} = C \\ X_{22}A = AX_{22} \end{matrix}

which, together with the observability of (A, C), imply that X_{22} = I. Setting X = −X_{21} and using the fact that ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) is symplectic we get X^{∗} = X. Moreover, we have now

B_{0} = B − XC^{∗}  (78)

and

XA^{∗} − BB^{∗} = −B_{0}B_{0}^{∗} − AX

or

XA^{∗} + AX − BB^{∗} + B_{0}B_{0}^{∗} = 0  (79)

Using (78) we have XA^{∗}+AX −BB^{∗}+(B−XC^{∗})(B^{∗}−CX) = 0 This can be rewritten as

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0.  (80)

2. As A − BC is stable, this homogeneous Ricatti equation has a unique positive definite solution X_{+}. This solution is such that A − BC + XC^{∗}C is antistable. Thus W_{+} = ( \begin{matrix}A & B − X_{+}C^{∗} \\ C & I \end{matrix} ). Necessarily, this is a minimal realization as W_{+}W_{+}^{∗} = Φ and δ(Φ) = 2n.

Next we compute a state space representation of the inner function Q_{+}. Proposition 6.1 Let W_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} ) be a minimal realization, and let X_{+} be the positive definite solution of the Riccati equation

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0.  (81)

Then Q_{+} = W_{−}^{−1}W_{+} has a minimal realization given by

Q_{+} = ( \begin{matrix}A − BC & −X_{+}C^{∗} \\ C & I \end{matrix} )  (82)

Proof: Since W_{−}^{−1} = ( \begin{matrix}A − BC & B \\ −C & I \end{matrix} ), we have

\begin{aligned}Q_{+} & = & ( \begin{matrix}A − BC & B \\ −C & I \end{matrix} ) × ( \begin{matrix}A & B − X_{+}C^{∗} \\ C & I \end{matrix} ) \\ & = & ( \begin{matrix}A & 0 & B − X_{+}C^{∗} \\ BC & A − BC & B \\ C & −C & I \end{matrix} ) \end{aligned}

We apply now the similarity ( \begin{matrix}I & I \\ 0 & I \end{matrix} ) to get

Q_{+} = ( \begin{matrix}A − BC & 0 & −X_{+}C^{∗} \\ BC & A & B \\ C & 0 & I \end{matrix} ) = I − C(sI − A + BC)^{−1}X_{+}C^{∗}

or (82) follows.

We claim that Q_{+} has McMillan degree n. It suffices to show that the previous realization is minimal. Observability is obvious. To see its reachability we rewrite (82) as

Q_{+} = ( \begin{matrix}X_{+}^{−1}(A − BC)X_{+} & −C^{∗} \\ CX_{+} & I \end{matrix} )

However, from the Riccati equation (81), we get

X_{+}^{−1}(A − BC)X_{+} = −A^{∗} + C^{∗}B^{∗} − C^{∗}CX_{+} = −A^{∗} + C^{∗}(B^{∗} − CX_{+})

i.e.

Q_{+} = ( \begin{matrix}−A^{∗} + C^{∗}(B^{∗} − CX_{+}) & −C^{∗} \\ CX_{+} & I \end{matrix} )

and the reachability of this realization clearly follows from the observability of (A, C).

We can check now that a minimal, stable spectral factor is of the form W = W_{−}Q for some left inner factor of Q_{+}. Indeed, by Theorem 4.3, a left factor of Q_{+} is given

by Q = ( \begin{matrix}A − BC & −XC^{∗} \\ C & I \end{matrix} ), where X is a nonnegative definite solution of the Riccati

equation (81). Therefore

W_{−}Q = ( \begin{matrix}A & B \\ C & I \end{matrix} ) × ( \begin{matrix}A − BC & −XC^{∗} \\ C & I \end{matrix} )

= ( \begin{matrix}A − BC & 0 & −XC^{∗} \\ BC & A & B \\ C & C & I \end{matrix} )

= ( \begin{matrix}A & B − XC^{∗} \\ C & I \end{matrix} )

Our next step is the computation of all minimal, maximum phase spectral factors. We do this first based on a state space realization of W_{+}. At the next stage we express the various spectral factors in terms of a state space realization of W_{−}.

Theorem 6.2 Let Φ be a coercive spectral function, normalized by Φ(∞) = I, and let

W_{+} = ( \begin{matrix}A & B \\ C & I \end{matrix} )  (83)

be a minimal realization of the stable, maximum phase spectral factor of Φ. Then

1. W is a minimal, maximum phase spectral factor of Φ if and only if

W = ( \begin{matrix}A + BB^{∗}Y & B \\ C + B^{∗}Y & I \end{matrix} )  (84)

where Y is a nonnegative definite solution of the homogeneous Riccati equation

A^{∗}Y + Y A + Y BB^{∗}Y = 0.  (85)

2. \overline{W}_{+} is the minimal, maximum phase, antistable spectral factor of Φ if and only if

\overline{W}_{+} = ( \begin{matrix}A + BB^{∗}Y_{−} & B \\ C + B^{∗}Y_{−} & I \end{matrix} )  (86)

where Y_{−} is the positive definite solution of the homogeneous Riccati equation (85).

Proof:

1. Let us assume W_{+} has the minimal realization (83). We assume that

W_{+} = E_{+}D_{−}^{−1}  (87)

is a polynomial, right coprime factorization with D_{−} stable and E_{+} antistable. Let W be an arbitrary, minimal, maximum phase spectral factor. Since we want to keep the zero structure we assume W = E_{+}D^{−1}. From the equality Φ = W_{+}W_{+}^{∗} = W W^{∗} we get

D^{∗}D = D_{−}^{∗}D_{−}  (88)

Thus D is determined from D_{−} by a process of spectral factorization. Comparing

\{ \begin{matrix}W_{+} & = & E_{+}D_{−}^{−1} \\ W & = & E_{+}D^{−1} \end{matrix}

it follows that

\{ \begin{matrix}W_{+}^{−1} & = & D_{−}E_{+}^{−1} \\ W^{−1} & = & DE_{+}^{−1} \end{matrix}

and hence, since

W_{+}^{−1} = ( \begin{matrix}A − BC & B \\ −C & I \end{matrix} )

we may assume that

W^{−1} = ( \begin{matrix}A − BC & B \\ −C_{0} & I \end{matrix} ) .

Thus, by inversion, we have

W = ( \begin{matrix}A − B(C − C_{0}) & B \\ C_{0} & I \end{matrix} )

and

W^{∗} = ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & C_{0}^{∗} \\ −B^{∗} & I \end{matrix} )

Using two cascade realizations of Φ we have

\begin{aligned}W_{+}W_{+}^{∗} & = & ( \begin{matrix}A & B \\ C & I \end{matrix} ) × ( \begin{matrix}−A^{∗} & C^{∗} \\ −B^{∗} & I \end{matrix} ) \\ & = & ( \begin{matrix}−A^{∗} & 0 & C^{∗} \\ −BB^{∗} & A & B \\ −B^{∗} & C & I \end{matrix} ) . \end{aligned}

Similarly,

\begin{aligned}W W^{∗} & = & ( \begin{matrix}A − B(C − C_{0}) & B \\ C_{0} & I \end{matrix} ) × ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & C_{0}^{∗} \\ −B^{∗} & I \end{matrix} ) \\ & = & ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & 0 & C_{0}^{∗} \\ −BB^{∗} & A − B(C − C_{0}) & B \\ −B^{∗} & C_{0} & I \end{matrix} ) . \end{aligned}

By the state space isomorphism theorem the two realizations are isomorphic. Moreover, the isomorphism ( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} ) is necessary symplectic. Thus we have the equalities

\{ \begin{matrix}( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} ) ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) ( \begin{matrix}Y_{11}^{∗} & Y_{21}^{∗} \\ Y_{12}^{∗} & Y_{22}^{∗} \end{matrix} ) & = & ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) \\ ( \begin{matrix}Y_{11}^{∗} & Y_{21}^{∗} \\ Y_{12}^{∗} & Y_{22}^{∗} \end{matrix} ) ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) ( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} ) & = & ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) \end{matrix}  (89)

First we will show that the previous isomorphism is necessarily of the form ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) with Y^{∗} = Y.

The isomorphism equations are

\{ \begin{matrix}( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} ) ( \begin{matrix}C^{∗} \\ B \end{matrix} ) = ( \begin{matrix}C_{0}^{∗} \\ B \end{matrix} ) \\ ( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} ) & ( \begin{matrix}−A^{∗} & 0 \\ −BB^{∗} & A \end{matrix} ) \\ = ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & 0 \\ −BB^{∗} & A − B(C − C_{0}) \end{matrix} ) ( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} ) \\ ( −B^{∗} C ) = ( −B^{∗} C_{0} ) ( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} ) \end{matrix}  (90)

These matrix equations can be rewritten as

\begin{matrix}Y_{11}C^{∗} + Y_{12}B = C_{0}^{∗} \\ Y_{21}C^{∗} + Y_{22}B = B \end{matrix}  (91)

\begin{matrix}−Y_{11}A^{∗} − Y_{12}BB^{∗} = [−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗}]Y_{11} \\ Y_{12}A = [−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗}]Y_{12} \\ −Y_{21}A^{∗} − Y_{22}BB^{∗} = −BB^{∗}Y_{11} + [A − B(C − C_{0})]Y_{21} \\ Y_{22}A = −BB^{∗}Y_{12} + [A − B(C − C_{0})]Y_{22} \end{matrix}  (92)

and

\begin{matrix}−B^{∗} = −B^{∗}Y_{11} + C_{0}Y_{21} \\ C = −B^{∗}Y_{12} + C_{0}Y_{22} \end{matrix}  (93)

Substituting Y_{22}B = B−Y_{21}C^{∗} into −Y_{21}A^{∗}−Y_{22}BB^{∗} = −BB^{∗}Y_{11}+[A−B(C−C_{0})]Y_{21} we get

− Y_{21}(A^{∗} − C^{∗}B^{∗}) = (A − BC)Y_{21}  (94)

Now, as

W_{+}^{−1} = ( \begin{matrix}A − BC & B \\ −C & I \end{matrix} )

it follows that A − BC is antistable. Therefore equation (94) implies that Y_{21} = 0.

Next we substitute Y_{12}B = C_{0}^{∗} − Y_{11}C^{∗} into the first equation of (92), and use the first equation of (93), to get

\{ \begin{matrix}B^{∗} & = & B^{∗}Y_{11} \\ (A^{∗} − C^{∗}B^{∗})Y_{11} & = & Y_{11}(A^{∗} − C^{∗}B^{∗}) \end{matrix}  (95)

This pair of equations implies of course that Y_{11} = I. From the fact that ( \begin{matrix}Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} )

is symplectic, using Y_{11} = I we deduce that Y_{12}^{∗} = Y_{12}. Next we use this in (91) to get C − C_{0} = −B^{∗}Y_{12}. Upon substitution in the last equation of (93) we get

\{ \begin{matrix}Y_{22}B & = & B \\ Y_{22}A & = & AY_{22} \end{matrix}  (96)

These two equations imply, using reachability of the pair (A, B), that Y_{22} = I.

Defining now Y = Y_{12}, the isomorphism equations simplify now to

\{ \begin{matrix}( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) ( \begin{matrix}C^{∗} \\ B \end{matrix} ) = ( \begin{matrix}C_{0}^{∗} \\ B \end{matrix} ) \\ ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) & ( \begin{matrix}−A^{∗} & 0 \\ −BB^{∗} & A \end{matrix} ) \\ = ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & 0 \\ −BB^{∗} & A − B(C − C_{0}) \end{matrix} ) ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) \\ ( −B^{∗} C ) = ( −B^{∗} C_{0} ) ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) \end{matrix}  (97)

Equating terms, we get

C_{0} = C − B^{∗}Y  (98)

and

Y A = −A^{∗}Y + (C^{∗} − C_{0}^{∗})B^{∗}Y

which, using (98), implies

A^{∗}Y + Y A + Y BB^{∗}Y = 0.  (99)

As A is stable, all symmetric solutions of this Riccati equation are nonnegative definite. This leads to the representation (84) of the spectral factor.

2. Rewriting the Riccati equation (99) as

(A^{∗} + Y BB^{∗})Y + Y (A + BB^{∗}Y) − Y BB^{∗}Y = 0.  (100)

it is clear that the positive definite solution Y_{−} is the destabilizing solution, namely the solution for which A + BB^{∗}Y is antistable.

We proceed to compute a state space representation of the inner function K_{+}.

Proposition 6.2 Let W_{+} = ( \begin{matrix}A & B \\ C & I \end{matrix} ) be a minimal realization, and let Y_{−} be the positive

definite solution of the Riccati equation

A^{∗}Y + Y A + Y BB^{∗}Y = 0.  (101)

Then

1. K_{+} = \overline{W} _{+}^{−1}W_{+} has a minimal realization given by

K_{+} = ( \begin{matrix}A & B \\ −B^{∗}Y_{−} & I \end{matrix} )  (102)

2. Let W be a minimal, maximum phase spectral factor having the minimal realization

W = ( \begin{matrix}A + BB^{∗}Y & B \\ C + B^{∗}Y & I \end{matrix} )  (103)

where Y is a nonnegative definite solution of the homogeneous Riccati equation (101). Then

K_{1} = \overline{W} _{+}^{−1}W = ( \begin{matrix}A + BB^{∗}Y & B \\ −B^{∗}(Y_{−} − Y) & I \end{matrix} )  (104)

and

K_{2} = W^{−1}W_{+} = ( \begin{matrix}A & B \\ −B^{∗}Y & I \end{matrix} )  (105)

Proof:

1. We have, using the similarity ( \begin{matrix}I & 0 \\ I & I \end{matrix} ),

K_{+} = \overline{W} _{+}^{−1}W_{+} = ( \begin{matrix}A − BC & B \\ −(C + B^{∗}Y_{−}) & I \end{matrix} ) × ( \begin{matrix}A & B \\ C & I \end{matrix} )

= ( \begin{matrix}A & 0 & B \\ BC & A − BC & B \\ C & −(C + B^{∗}Y_{−}) & I \end{matrix} ) = ( \begin{matrix}A & 0 & B \\ 0 & A − BC & 0 \\ −B^{∗}Y_{−} & −(C + B^{∗}Y_{−}) & I \end{matrix} )

= ( \begin{matrix}A & B \\ −B^{∗}Y_{−} & I \end{matrix} )

2. The proof is similar.

Note that when Y = 0 we have K_{1} = K_{+} and K_{2} = I whereas, when Y = Y_{−} we have K_{1} = I and K_{2} = K_{+}. In general the two realizations (104) and (105) are not minimal. We note however that rank(Y_{−} − Y) + rankY = rankY_{−}.

While the previous results give state space realizations in terms of W_{+} we would really like to have the results stated in terms of a minimal realization of the minimum phase, stable spectral factor. To this end we would like to represent Y_{−} as a function of W_{−} and the positive definite solution X_{+} of the Riccati equation (76).

Proposition 6.3 Let

W_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} )  (106)

be a minimal realization of the stable, minimum phase spectral factor of Φ. Let

W_{+} = ( \begin{matrix}A & B \\ C & I \end{matrix} ) = ( \begin{matrix}A & B − X_{+}C^{∗} \\ C & I \end{matrix} )  (107)

where X_{+} is the positive definite solution of the homogeneous Riccati equation

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0.  (108)

and let Z be the positive definite solution of the Riccati equation

A^{∗}Z + ZA + ZBB^{∗}Z = 0.  (109)

Then, if Y_{−} is the positive definite solution of the Riccati equation

A^{∗}Y + Y A + Y BB^{∗}Y = 0  (110)

then we have

Y_{−} = (I + ZX)^{−1}Z = Z(I + XZ)^{−1}.  (111)

Proof: Considering the realization (107), the Riccati equation (110) can be rewritten as

\begin{aligned}0 & = & A^{∗}Y + Y A + Y (B − X_{+}C^{∗})(B^{∗} − CX_{+}) \\ & = & A^{∗}Y + Y A + Y BB^{∗}Y − Y X_{+}C^{∗}BY − Y B^{∗}CXY + Y XCC^{∗}XY \end{aligned}

On the other hand, multiplying equation (108) on both sides by Y, we get

Y X(A^{∗} − C^{∗}B^{∗})Y + Y (A − BC)XY + Y XC^{∗}CXY = 0.

Subtracting these two equations we obtain

(I − Y X)A^{∗}Y + Y A(I − XY) + Y BB^{∗}Y = 0

Multiplying this equation on the left and right sides by (I − Y X)^{−1} and (I − XY)^{−1} respectively, we get

A^{∗}Y (I − XY)^{−1} + (I − Y X)^{−1}Y A + (I − Y X)^{−1}Y BB^{∗}Y (I − XY)^{−1} = 0

Comparing this with equation (109) we conclude that

Z = Y (I − XY)^{−1}

which implies (111).

Corollary 6.1 Let

W_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} )  (112)

be a minimal realization of the stable, minimum phase spectral factor of Φ. Let X_{+} be the positive definite solution of the homogeneous Riccati equation

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0.  (113)

and let Z be the positive definite solution of the Riccati equation

A^{∗}Z + ZA + ZBB^{∗}Z = 0.  (114)

Then

\overline{W}_{+} = ( \begin{matrix}A + (B − X_{+}C^{∗})(B^{∗} − CX_{+})(I + ZX_{+})^{−1}Z & B − X_{+}C^{∗} \\ C + (B^{∗} − CX_{+})(I + ZX_{+})^{−1}Z & I \end{matrix} )  (115)

and

K_{+} = ( \begin{matrix}A & B \\ −B^{∗}(I + ZX_{+})^{−1}Z & I \end{matrix} )  (116)

It is worthwhile to check that the generator matrix for \overline{W} _{+}^{−1} is A − BC + X_{+}CC^{∗} which is antistable as expected.

Of course the transformation from W_{−} to \overline{W}_{+} can be achieved by flipping poles first and zeroes later. Since the proofs are similar to those in the first part of this section, we are going to omit them and content ourselves with stating the results.

Theorem 6.3 Let Φ be a coercive spectral function, normalized by Φ(∞) = I, and let

W_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} )  (117)

be a minimal realization of the stable, minimum phase spectral factor of Φ. Then

1. W is a minimal, minimum phase spectral factor of Φ if and only if

W = ( \begin{matrix}A + BB^{∗}Z & B \\ C + B^{∗}Z & I \end{matrix} )  (118)

where Z is a nonnegative definite solution of the homogeneous Riccati equation

A^{∗}Z + ZA + ZBB^{∗}Z = 0.  (119)

2. A state space realization of the minimum phase, antistable spectral factors is given by

\overline{W}_{−} = ( \begin{matrix}A + BB^{∗}Z_{+} & B \\ C + B^{∗}Z_{+} & I \end{matrix} )  (120)

where Z_{+} is the positive definite solution of the homogeneous Riccati equation (119). This is the solution for which A + BB^{∗}Z_{+} is antistable.

Theorem 6.4 Let Φ be a coercive spectral function, normalized by Φ(∞) = I, and let

\overline{W}_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} )  (121)

be a minimal realization of the antistable, minimum phase spectral factor of Φ. Then

1. W is a minimal, antistable spectral factor of Φ if and only if

W = ( \begin{matrix}A & B − XC^{∗} \\ C & I \end{matrix} )  (122)

where X is a nonnegative definite solution of the homogeneous Riccati equation

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0.  (123)

2. A state space realization of the maximum phase, antistable spectral factors is given by

\overline{W}_{+} = ( \begin{matrix}A & B − X_{+}C^{∗} \\ C & I \end{matrix} )  (124)

where X_{+} is the positive definite solution of the homogeneous Riccati equation (123).

7 State space computation of Lindquist-Picci pairs

In this section we present state space formulas for the computation of primal and dual Lindquist-Picci pairs of a given, minimal spectral factor. The approach to these computations uses the connection, established in section 3, between LP pairs and DSS factorizations. To this we add the state space formulas for the computation of DSS factorizations derived in section 5. Thus we can state the following.

Theorem 7.1 Let W_{1} = ( \begin{matrix}A & B \\ C & I \end{matrix} ) be a minimal realization of a minimal spectral factor of

the spectral function Φ. Then

1. Let (W, \overline{W}) be a minimal Lindquist Picci pair for W_{1}. Then state space representation of W and \overline{W} are given by

(a)

W = ( \begin{matrix}A − BB^{∗}X & B \\ C − B^{∗}X & I \end{matrix} )  (125)

where X is the unique, stabilizing solution, i.e. the solution for which A − BB^{∗}X is stable, of the homogeneous Riccati equation

A^{∗}X + XA − XBB^{∗}X = 0.  (126)

(b)

\overline{W} = ( \begin{matrix}A + BB^{∗}Y & B \\ C + B^{∗}Y & I \end{matrix} )  (127)

where Y is the unique, antistabilizing solution, i.e. A + BB^{∗}Y is antistable, of the homogeneous Riccati equation

A^{∗}Y + Y A + Y BB^{∗}Y = 0  (128)

i.e. the solution that makes A + BB^{∗}Y antistable.

2. Let (W_{δ}, \overline{W}_{δ}) be the dual Lindquist Picci pair for W_{1}. Then state space representation of W_{δ} and \overline{W}_{δ} are given by

(a)

W_{δ} = ( \begin{matrix}A & B + ZC^{∗} \\ C & I \end{matrix} )  (129)

where Z is the unique, stabilizing solution, i.e. the solution for which A − BC − ZC^{∗}C is stable, of the homogeneous Riccati equation

Z(A^{∗} − C^{∗}B^{∗}) + (A − BC)Z = ZC^{∗}CZ.  (130)

(b)

\overline{W}_{δ} = ( \begin{matrix}A & B − ZC^{∗} \\ C & I \end{matrix} )  (131)

where Z is the unique, antistabilizing solution, i.e. A − BC + ZC^{∗}C is antistable, of the homogeneous Riccati equation

Z(A^{∗} − C^{∗}B^{∗}) + (A − BC)Z = −ZC^{∗}CZ.  (132)

3. (a) The inner function V = W_{1}^{−1}W has a realization

V = ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \end{matrix} )  (133)

Moreover we have

δ(V) = n − dim ker X  (134)

In fact ker X is the unobservable space of the realization (133).

(b) The inner function U = \overline{W}^{−1}W_{1} has a realization

U = ( \begin{matrix}A & B \\ −B^{∗}Y & I \end{matrix} )  (135)

Moreover we have

δ(U) = n − dim ker Y  (136)

In fact ker Y is the unobservable space of the realization (127).

Proof:

1. We apply throughout Proposition 3.2 and Theorem 5.1.

(a) The numerator N_{1} of a right DSS factorization is given by the state space representation (125) where X is the unique, stabilizing solution of the Riccati equation (126).

(b) To compute \overline{W} we have first to compute the numerator of a left DSS factoriztion of W_{1}^{∗}. Now

W_{1}^{∗} = ( \begin{matrix}−A^{∗} & C^{∗} \\ −B^{∗} & I \end{matrix} ) = ( \begin{matrix}A & B \\ C & I \end{matrix} ) .

This is given by

N_{2} = ( \begin{matrix}A − ZC^{∗}C & B − ZC^{∗} \\ C & I \end{matrix} )  (137)

where Z is the unique, stabilizing solution of the algebraic Riccati equation

ZA^{∗} + AZ = ZC^{∗}CZ.  (138)

This is equivalent to the equation

A^{∗}Z + ZA + ZBB^{∗}Z = 0.  (139)

So

N_{2} = ( \begin{matrix}−A^{∗} − ZBB^{∗} & C^{∗} + ZB \\ −B^{∗} & I \end{matrix} )  (140)

Hence

\overline{W} = N_{2}^{∗} = ( \begin{matrix}A + BB^{∗}Z & B \\ C + B^{∗}Z & I \end{matrix} )  (141)

(a) To compute W_{δ} we have first to compute the numerator of a left DSS factoriztion of W_{1}^{−1}. Now

W_{1}^{−1} = ( \begin{matrix}A − BC & B \\ −C & I \end{matrix} ) = ( \begin{matrix}A & B \\ C & I \end{matrix} ) .

So

N_{3} = ( \begin{matrix}A − ZC^{∗}C & B − ZC^{∗} \\ C & I \end{matrix} ) .

Here Z is the stabilizing solution of the Riccati equation

ZA^{∗} + AZ = ZC^{∗}CZ.  (142)

which can be rewritten as

Z(A^{∗} − C^{∗}B^{∗}) + (A − BC)Z = ZC^{∗}CZ.  (143)

So

\begin{aligned}N_{3}^{−1} & = & ( \begin{matrix}A − BC & B − ZC^{∗} \\ −C & I \end{matrix} ) \\ & = & ( \begin{matrix}A & B + ZC^{∗} \\ C & I \end{matrix} ) = W_{δ} \end{aligned}

This implies realization (129).

(b) Finally, to compute \overline{W}_{δ}, we neen to compute first the numerator of aright DSS factorization of W_{1}^{−∗}. Now

W_{1}^{−∗} = ( \begin{matrix}−A^{∗} + C^{∗}B^{∗} & C^{∗} \\ B^{∗} & I \end{matrix} ) = ( \begin{matrix}A & B \\ C & I \end{matrix} ) .

So

N_{4} = ( \begin{matrix}A − BB^{∗}X & B \\ C − B^{∗}X & I \end{matrix} ) .

Here X is the stabilizing solution of the Riccati equation

A^{∗}X + XA = XBB^{∗}X.  (144)

which can be rewritten as

(A − BC)X + X(A^{∗} − C^{∗}B^{∗}) = XC^{∗}CX.  (145)

Thus

N_{4} = ( \begin{matrix}−A^{∗} + C^{∗}B^{∗} − C^{∗}CX^{∗} & C^{∗} \\ B^{∗} − CX & I \end{matrix} )

which, by taking adjoints, implies

N_{4}^{∗} = ( \begin{matrix}A − BC + XC^{∗}C & B − XC^{∗} \\ −C & I \end{matrix} )

We conclude that

\overline{W}_{δ} = N_{4}^{−∗} = ( \begin{matrix}A & B − XC^{∗} \\ C & I \end{matrix} ) .

3.

(a) We compute

\begin{aligned}V & = & W_{1}^{−1}W = ( \begin{matrix}A − BC & B \\ −C & I \end{matrix} ) × ( \begin{matrix}A − BB^{∗}X & B \\ C − B^{∗}X & I \end{matrix} ) \\ & = & ( \begin{matrix}A − BB^{∗}X & 0 & B \\ B(C − B^{∗}X) & A − BC & B \\ C − B^{∗}X & −C & I \end{matrix} ) \end{aligned}

Since

( \begin{matrix}I & 0 \\ −I & I \end{matrix} ) ( \begin{matrix}B \\ B \end{matrix} ) = ( \begin{matrix}B \\ 0 \end{matrix} )

( \begin{matrix}I & 0 \\ −I & I \end{matrix} ) ( \begin{matrix}A − BB^{∗}X & 0 \\ B(C − B^{∗}X) & A − BC \end{matrix} ) ( \begin{matrix}I & 0 \\ I & I \end{matrix} )

= ( \begin{matrix}A − BB^{∗}X & 0 \\ −A + BC & A − BC \end{matrix} ) ( \begin{matrix}I & 0 \\ I & I \end{matrix} )

= ( \begin{matrix}A − BB^{∗}X & 0 \\ 0 & A − BC \end{matrix} )

( C − B^{∗}X −C ) ( \begin{matrix}I & 0 \\ I & I \end{matrix} ) = ( −B^{∗}X −C )

So

V = ( \begin{matrix}A − BB^{∗}X & 0 & B \\ 0 & A − BC & 0 \\ −B^{∗}X & −C & I \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \end{matrix} )

which proves (133).

Now the realization (133) is clearly reachable. So this would be a minimal realization if and only if it is observable. Thus we proceed to compute the unobservable space of this realization. The Riccati equation (126) can be rewritten as

A^{∗}X = −X(A − BB^{∗}X)

This equality implies X(A − BB^{∗}X)^{i} = (−1)^{i}(A^{∗})^{i}X and hence

B^{∗}X(A − BB^{∗}X)^{i} = (−1)^{i}B^{∗}(A^{∗})^{i}X

This shows that

∩_{i≥0} ker B^{∗}X(A − BB^{∗}X)^{i} = ∩_{i≥0} ker B^{∗}(A^{∗})^{i}X

By the reachability of the pair (A, B), it follows that

ker X = ∩_{i≥0} ker B^{∗}X(A − BB^{∗}X)^{i}.  (146)

(b) We compute

\begin{aligned}U & = & \overline{W}^{−1}W_{1} = ( \begin{matrix}A − BC & B \\ −(C + B^{∗}Y) & I \end{matrix} ) × ( \begin{matrix}A & B \\ C & I \end{matrix} ) \\ & = & ( \begin{matrix}A & 0 & B \\ BC & A − BC & B \\ C & −(C − B^{∗}Y) & I \end{matrix} ) \end{aligned}

As

\{ \begin{matrix}( \begin{matrix}I & 0 \\ −I & I \end{matrix} ) ( \begin{matrix}B \\ B \end{matrix} ) = ( \begin{matrix}B \\ 0 \end{matrix} ) \\ ( \begin{matrix}I & 0 \\ −I & I \end{matrix} ) ( \begin{matrix}A & 0 \\ BC & A − BC \end{matrix} ) ( \begin{matrix}I & 0 \\ I & I \end{matrix} ) = ( \begin{matrix}A & 0 \\ 0 & A − BC \end{matrix} ) \\ ( C −(C + B^{∗}Y) ) ( \begin{matrix}I & 0 \\ I & I \end{matrix} ) = ( −B^{∗}Y −(C + B^{∗}Y) ) \end{matrix}

So it follows that

U = ( \begin{matrix}A & 0 & B \\ 0 & A − BC & 0 \\ −B^{∗}Y & −(C + B^{∗}Y) & I \end{matrix} ) = ( \begin{matrix}A & B \\ −B^{∗}Y & I \end{matrix} )

That

ker Y = ∩_{i≥0} ker B^{∗}X(A − BB^{∗}X)^{i}.  (147)

It is instructive to compare equations (127) and (131) with (118) and (75) respectively.

8 Parametrization of minimal spectral factors

We can apply now the results and techniques developed in the preceeding sections to obtain a complete parametrization of all minimal spectral factors. Now a minimal spectral factor is completely determined by a primal and dual Lindquist-Picci pairs, taken together, and these pairs are in turn completely determined by factorizations of the inner functions Q_{+} and K_{−}. We saw in section 4 that the factorizations of inner functions are completely determined by solutions of associated Riccati equations. Thus we are led to the parametrization of the set of minimal spectral factors in terms of state space formulas using two solutions of Riccati equations. This is done in the next theorem.

Theorem 8.1 Let W_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} ) be a minimal realization of the stable, minimum phase

spectral factor of a normalized spectral function Φ. Let X be a nonnegative definite solution of the Riccati equation

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0,  (148)

and let Z be a nonnegative definite solution of the Riccati equation

A^{∗}Z + ZA + ZBB^{∗}Z = 0.  (149)

Then each of the following realizations

W_{1} = ( \begin{matrix}A + (B − XC^{∗})(B^{∗} − CX)Z(I + XZ)^{−1} & B − XC^{∗} \\ C + (B^{∗} − CX)Z(I + XZ)^{−1} & I \end{matrix} )  (150)

and

W_{1} = ( \begin{matrix}A + BB^{∗}Z & B − (I + XZ)^{−1}X(C^{∗} + ZB) \\ C + B^{∗}Z & I \end{matrix} )  (151)

gives a complete parametrization of all minimal spectral factors of Φ.

The realizations (151) and (150) are isomorphic and the intertwining isomorphism is given by the map I + XZ.

Proof: Let W_{1} be a minimal spectral factor. Let (W, \overline{W}) be the Lindquist-Picci pair determined by W_{1}. By Theorem 6.1, W has a state space representation of the form

W = ( \begin{matrix}A & B − XC^{∗} \\ C & I \end{matrix} ) = ( \begin{matrix}A & B \\ C & I \end{matrix} ) where X is a nonnegative definite solution of the Ric-

cati equation (148). Assume we have the polynomial left coprime factorization W = ED_{−}^{−1} with D_{−} necessarily a stable polynomial matrix. Since W_{1} differs from W in its pole structure only it follows that it must have a left coprime factorization W_{1} = ED^{−1}. We compare now

\begin{aligned}W^{−1} & = & D_{−}E^{−1} \\ W_{1}^{−1} & = & DE^{−1} \end{aligned}

noting that W^{−1} = ( \begin{matrix}A − BC & B \\ −C & I \end{matrix} ), we infer that W_{1}^{−1} = ( \begin{matrix}A − BC & B \\ −C_{0} & I \end{matrix} ). Since

\begin{aligned}W W^{∗} & = & ( \begin{matrix}A & B \\ C & I \end{matrix} ) × ( \begin{matrix}−A^{∗} & C^{∗} \\ −B^{∗} & I \end{matrix} ) \\ & = & ( \begin{matrix}−A^{∗} & 0 & C^{∗} \\ −BB^{∗} & A & B \\ −B^{∗} & C & I \end{matrix} ) \end{aligned}

Now from our representation of W_{1}^{−1} we conclude that W_{1} = ( \begin{matrix}A − B(C − C_{0}) & B \\ C_{0} & I \end{matrix} ). There-

fore

\begin{aligned}W_{1}W_{1}^{∗} & = & ( \begin{matrix}A − B(C − C_{0}) & B \\ C_{0} & I \end{matrix} ) × ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & C_{0}^{∗} \\ −B^{∗} & I \end{matrix} ) \\ & = & ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & 0 & C_{0}^{∗} \\ −BB^{∗} & A − B(C − C_{0}) & B \\ −B^{∗} & C_{0} & I \end{matrix} ) \end{aligned}

Using previous arguments based on the state space isomorphism theorem, there exists a Hermitian Y such that

\begin{matrix}( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) ( \begin{matrix}C^{∗} \\ B \end{matrix} ) = ( \begin{matrix}C_{0}^{∗} \\ B \end{matrix} ) \\ ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) ( \begin{matrix}−A^{∗} & 0 \\ −BB^{∗} & A \end{matrix} ) = ( \begin{matrix}−A^{∗} + (C^{∗} − C_{0}^{∗})B^{∗} & 0 \\ −BB^{∗} & A − B(C − C_{0}) \end{matrix} ) ( \begin{matrix}I & Y \\ 0 & I \end{matrix} ) \end{matrix}

These equations lead to

\{ \begin{matrix}C_{0}^{∗} = C^{∗} + Y B \\ Y A + A^{∗}Y + Y BB^{∗}Y = 0 \end{matrix}  (152)

and Y is a nonnegative definite solution of this Riccati equation. Since ( \begin{matrix}A & B \\ C & I \end{matrix} ) =

( \begin{matrix}A & B − XC^{∗} \\ C & I \end{matrix} ) the Riccati equation can be rewritten in terms of the W_{−} data as

Y A + A^{∗}Y + Y (B − XC^{∗})(B^{∗} − CX)Y = 0  (153)

and so

W_{1} = ( \begin{matrix}A + (B − XC^{∗})(B^{∗} − CX)Y & B − XC^{∗} \\ C + (B^{∗} − CX)Y & I \end{matrix} )  (154)

While this provides an answer to a state space representation of W_{1}, we would like to get the answer in terms of solutions of the two Riccati equations, namely (148) and (149), associated with W_{−}. To this end we multiply equation (148) on both sides by Y and subtract from the Riccati equation (153) to get

0 = Y A + A^{∗}Y + Y (B − XC^{∗})(B^{∗} − CX)Y
−Y [X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX]Y
= (I − Y X)A^{∗}Y = Y A(I − XY) + Y BB^{∗}Y

This implies that

Z = Y (I − XY)^{−1}  (155)

is a, necessarily nonnegative definite, solution of the Riccati equation (149). But this last equation implies

Y = (I + ZX)^{−1}Z = Z(I + XZ)^{−1}.  (156)

Thus we get the parametrization (150) in terms of solutions to the Riccati equations (148) and (149).

Here we follow the alternative path. Let (W′, \overline{W}′) be the dual Lindquist-Picci pair determined by W_{1}. By Theorem 6.3, W′ has a state space representation of the form

W′ = ( \begin{matrix}A + BB^{∗}Z & B \\ C + B^{∗}Z & I \end{matrix} ) = ( \begin{matrix}A & B \\ C & I \end{matrix} )  (157)

where Z is a nonnegative definite solution of the Riccati equation (149). Since W′ and W_{1} have the same pole structure and differ only in their zero structure, we conclude by our

reasoning that W_{1} = ( \begin{matrix}A & B_{0} \\ C & I \end{matrix} ). Now

\begin{aligned}W′W^{′∗} & = & ( \begin{matrix}A & B \\ C & I \end{matrix} ) × ( \begin{matrix}−A^{∗} & C^{∗} \\ −B^{∗} & I \end{matrix} ) \\ & = & ( \begin{matrix}−A^{∗} & 0 & C^{∗} \\ −BB^{∗} & A & B \\ −B^{∗} & C & I \end{matrix} ) \end{aligned}

and

\begin{aligned}W_{1}W_{1}^{∗} & = & ( \begin{matrix}A & B_{0} \\ C & I \end{matrix} ) × ( \begin{matrix}−A^{∗} & C^{∗} \\ −B_{0}^{∗} & I \end{matrix} ) \\ & = & ( \begin{matrix}−A^{∗} & 0 & C^{∗} \\ −B_{0}B_{0}^{∗} & A & B_{0} \\ −B_{0}^{∗} & C & I \end{matrix} ) \end{aligned}

As W′W^{′∗} = W_{1}W_{1}^{∗} we are led to the isomorphism equations

\begin{matrix}( \begin{matrix}I & 0 \\ −U & I \end{matrix} ) ( \begin{matrix}C^{∗} \\ B \end{matrix} ) = ( \begin{matrix}C^{∗} \\ B_{0} \end{matrix} ) \\ ( \begin{matrix}I & 0 \\ −U & I \end{matrix} ) ( \begin{matrix}−A^{∗} & 0 \\ −BB^{∗} & A \end{matrix} ) = ( \begin{matrix}−A^{∗} & 0 \\ −B_{0}B_{0}^{∗} & A \end{matrix} ) ( \begin{matrix}I & 0 \\ −U & I \end{matrix} ) \end{matrix}

This leads to the equalities

\{ \begin{matrix}B_{0} = B − UC^{∗} \\ UA^{∗} + AU + B_{0}B_{0}^{∗} − BB^{∗} = 0 \end{matrix}

Using equation (157), we can rewrite the last Riccati equation as

0 = UA^{∗} + AU + B_{0}B_{0}^{∗} − BB^{∗}
= U(A^{∗} + ZBB^{∗}) + (A + BB^{∗}Z)U + (B − U(C^{∗} + ZB))(B^{∗} − (C + B^{∗}Z)U) − BB^{∗}
= U(A^{∗} + ZBB^{∗}) + (A + BB^{∗}Z)U − U(C^{∗} + ZB)B^{∗} − B(C + B^{∗}Z)U + U(C^{∗} + ZB)(C + B^{∗}Z)U
= U(A^{∗} − C^{∗}B^{∗}) + (A − BC)U + U(C^{∗} + ZB)(C + B^{∗}Z)U

Thus we get the state space representation

W_{1} = ( \begin{matrix}A + BB^{∗}Z & B − U(C^{∗} + ZB) \\ C + B^{∗}Z & I \end{matrix} )  (158)

Again we would like to express this realization in terms of solutions to the Riccati equations

(148) and (149).

To this end we compute, using A^{∗}Z + ZA + ZBB^{∗}Z = 0,

0 = U(A^{∗} − C^{∗}B^{∗}) + (A − BC)U + U(C^{∗} + ZB)(C + B^{∗}Z)U
= U(A^{∗} − C^{∗}B^{∗}) + (A − BC)U + UC^{∗}CU + UZBCU + UC^{∗}B^{∗}ZU + UZBB^{∗}ZU
= U(A^{∗} − C^{∗}B^{∗}) + (A − BC)U + UC^{∗}CU + UZBCU + UC^{∗}B^{∗}ZU + UZ(−ZA − A^{∗}Z)U
= U(A^{∗} − C^{∗}B^{∗})(I − ZU) + (I − UZ)(A − BC)U + UC^{∗}CU

This implies

(A − BC)U(I − ZU)^{−1} + (I − UZ)^{−1}U(A^{∗} − C^{∗}B^{∗}) + (I − UZ)^{−1}UC^{∗}CU(I − ZU)^{−1} = 0.

Setting X = U(I − ZU)^{−1} we see that X solves the Riccati equation (148). Solving for U, we get U = (I + XZ)^{−1}X = X(I + ZX)^{−1}, and this leads to the realization (151).

It remains to check whether the realizations (151) and (150) represent the same transfer function. It suffices to show that the realizations are isomorphic. In fact we will show that the map I + XZ provides the required isomorphism. This is shown by the following computations

(I + XZ)[B − (I + XZ)^{−1}X(C^{∗} + ZB)] = (I + XZ)B − X(C^{∗} + ZB) = B − XC^{∗},

[C + (B^{∗} − CX)Z(I + XZ)^{−1}](I + XZ) = C(I + XZ) + (B^{∗} − CX)Z = C + B^{∗}Z,

and, using the Riccati equations (148) and (149),

[A + (B − XC^{∗})(B^{∗} − CX)Z(I + XZ)^{−1}](I + XZ) − (I + XZ)(A + BB^{∗}Z) = A + AXZ + BB^{∗}Z − BCXZ − XC^{∗}B^{∗}Z + XC^{∗}CXZ − A − BB^{∗}Z − XZA − XZBB^{∗}Z = (A − BC)XZ − (XC^{∗}B^{∗} − XC^{∗}CX)Z − X[ZA + ZBBZ] = (A − BC)XZ + (XC^{∗}CX − XC^{∗}B^{∗})Z + XA^{∗}Z = (A − BC)XZ + X(A^{∗} − C^{∗}B^{∗})Z + XC^{∗}CXZ = [(A − BC)X + X(A^{∗} − C^{∗}B^{∗}) + XC^{∗}CX]Z = 0.

References

[1979] A. C. Antoulas, ”A polynomial matrix approach to F modG-invariant subspaces”, Doctoral Dissertation, Dept. of Mathematics, ETH Zurich.

[1949] A. Beurling, ”On two problems concerning linear transformations in Hilbert space”, Acta Math., 81, pp. 239-255.

[1972] R.W. Brockett and A. Rahimi, ”Lie algebras and linear differential equations”, in L. Weiss, Ed. Ordinary Differential Equations, Academic Press, New York.

[1958] M.S. Brodskii and M.S. Livsic, ”Spectral analysis of non-self-adjoint operators and intermediate systems”, Uspehi Math. Nauk. 13, 1-85; Amer. Math. Soc. Transl. Ser 2, 13, 265-346.

[1985] F.M. Callier, ”On polynomial matrix spectral factorization by symmetric root extraction”, IEEE Trans. Autom. Contr. AC-30, 453-464.

[1975] F.M. Callier and C.D. Nahum, ”Necessary and sufficient conditions for the complete controllability and observability of systems in series using the coprime decomposition of a rational matrix”, IEEE Trans. Circuits and Systems 22, 90-95.

[1974a] W. A. Coppel, ”Matrix quadratic equations”, Bull. Austr. Math. Soc., 10,377-401.

[1974b] W. A. Coppel, ”Matrices of rational functions”, Bull. Austral. Math. Soc. 11,89-113.

[1970] L. Debranges, ”Factorization and invariant subspaces”, J. Math. Anal. Appl. 29, 163-200.

[1971] R.G. Douglas, H.S. Shapiro & A.L. Shields, ”Cyclic vectors and invariant subspaces for the backward shift”, Ann. Inst. Fourier, Grenoble 20, 37-76.

[1985] H. Dym and N. Young, ””, Int. Eq. and Oper. Theor..

[1915] L. Fejér, ”Úber trigonometrische Polynome”, J. Reine und Angew. Math., 146, 53-82.

[1993] A. Ferrante, ”A parametrization of the minimal square spectral factors of a nonrational spectral density”, Preprint.

[1993] A. Ferrante, G. Michaletzky and M. Pavon, ”Parametrization of all minimal square spectral factors”, Preprint.

[1982] L. Finesso and G. Picci, ”A characterization of minimal spectral factors”, IEEE Trans. Autom. Contr. AC-27, 122-127.

[1968a] P. A. Fuhrmann, ”On the corona problem and its application to spectral problems in Hilbert space”, Trans. Amer. Math. Soc. 132, 55-67.

[1968b] P. A. Fuhrmann, ”A functional calculus in Hilbert space based on operator valued analytic functions”, Israel J. Math. 6, 267-278.

[1975] P. A. Fuhrmann, ”On Hankel operator ranges, meromorphic pseudo-continuation and factorization of operator valued analytic functions”, J. Lon. Math. Soc. (2) 13, 323-327.

[1976a] P. A. Fuhrmann, ”On series and parallel coupling of a class of discrete time infinite dimensional linear systems”, SIAM J. Contr. Optim., 14, 339-358.

[1976b] P. A. Fuhrmann, ”Algebraic system theory: An analyst’s point of view”, J. Franklin Inst., 301, 521-540.

[1977] P. A. Fuhrmann, ”On strict system equivalence and similarity”, Int. J. Contr. 25, 5-10.

[1983] P. A. Fuhrmann, ”On symmetric rational matrix functions”, Lin. Alg. Appl., Special issue on Linear Control Theory, 167-250.

[1981] P. A. Fuhrmann, Linear Systems and Operators in Hilbert Space, McGraw-Hill, New York.

[1983] P. A. Fuhrmann, ”On symmetric rational transfer functions”, Linear Algebra and Appl., 50,167-250.

[1985] P. A. Fuhrmann, ”The algebraic Riccati equation - a polynomial approach”, Systems and Control Letters, 369-376.

[1989] P. A. Fuhrmann, ”Elements of Factorization Theory from a Polynomial Point of View”, in H. Nijmeijer, J. M. S. Schumacher (eds.). Three Decades of Mathematical System Theory. A Collection of Surveys at the Occasion of the Fiftieth Birthday of Jan C. Willems. Lecture Notes in Control and Information Sciences Vol. 135. Springer Verlag, Berlin.

[1993] P. A. Fuhrmann and R. Ober, ”State space formulas for coprime factorizations”, T. Ando Festschrift, Birkhauser Verlag.

[1980] P. A. Fuhrmann and J.C. Willems,”A study of (A, B)-invariant subspaces via polynomial models”, Int. J. Contr. 31, 467-494.

[1982] I. Gohberg, P. Lancaster and L. Rodman, ”Factorization of selfadjoint matrix polynomials with constant signature”, Lin. and Multilin. Alg., 11, 209-224.

[1961] P.R. Halmos, ”Shifts on Hilbert spaces”, J. Reine Angew. Math., 208, 102-112.

[1978] M. L. J. Hautus and M. Heymann, ”Linear feedback - an algebraic approach”, SIAM J. Control 16,83-105.

REFERENCES 54 [1964] H. Helson, Lectures on Invariant Subspaces, Academic Press.

[1962] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall.

[1970] V.A. Jakubovic, ”Factorization of symmetric matrix polynomials”, Soviet Math. Doklady, 11, 1261-1264.

[1974] T. Kailath, ”A view of three decades of linear filtering theory”, IEEE Trans. Info.. Theo. IT-20, 146-181.

[1960] R.E. Kalman, ”Contributions to the theory of optimal control”, Bol. Soc. Math. Mexicana, 5, 102-119.

[1959] P.D. Lax, ”Translation invariant subspaces”, Acta Math., 101, 163-178.

[1979] A. Lindquist and G. Picci, ”On the stochastic realization problem”, SIAM J. Contr. Optimiz., 17, 365-389.

[1985] A. Lindquist and G. Picci, ”Realization theory for multivariable stationary Gaussian processes”, SIAM J. Contr. Optimiz., 23, 809-857.

[1993] R. Ober, ”Balanced parametrization of classes of linear systems”.

[1993] G. Picci and S. Pinzoni, ”Acausal models of stationary processes”, Preprint.

[1967] D. Sarason, ”Generalized interpolation in H^{∞}”, Trans. Amer. Math. Soc. 127, 179-203.

[1983] M.A. Shayman, ”Geometry of the algebraic Riccati equation”, Part I., SIAM J. Contr. Optim., 21, 375-394.

[1970] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam.

[1971] J.C. Willems,”Least squares stationary optimal control and the algebraic Riccati equation”, IEEE Trans. Autom. Contr. AC-16, 621-634.

[1979] H. Wimmer, ”A Jordan factorization theorem for polynomial matrices”, Proc. Amer. Math. Soc. 75, 201-206.

[1978] W.A. Wolovich, ”Skew prime polynomial matrices”, IEEE Trans. Autom. Contr. AC-23, 880-887.

[1961] D.C. Youla, ”On the factorization of rational matrices”, IRE Trans. Inf. Theor., IT-7, 172-189.