THE BOUNDED REAL CHARACTERISTIC FUNCTION AND NEHARI EXTENSIONS

P. A. Fuhrmann^{∗}^{†}

Dedicated to Professor M.S. Livsic

For the class of bounded real functions in the open right half plane we define a map, we call it the B-characteristic, which associates with each bounded real unction a stable function. The definition is based on normalized coprime factorizations of bounded real functions, the normalization being with respect to an indefinite metric. The construction bears great similarity to the characteristic functions studied in Fuhrmann and Ober [1993] for other classes of functions. We study the properties of the B-characteristic, the inverse map and the connections to balancing. We apply the construction to shed some new light on the problem of suboptimal Nehari complementation.

1 Introduction

In Fuhrmann and Ober [1993] two types of characteristic functions, the S-characteristic for the class of antistable transfer functions bounded in the left half plane and the L-charactristic for the class of minimal systems, were introduced. These characteristic functions played a central role in the development of a theory clarifying the duality between problems of model reduction on the one hand and robust control on the other. The L-characteristic, though not under this name, has initially been used by Glover and McFarlane [1989], see also Georgiou and Smith [1990], for characterizing the optimally robust stability margin for normalized coprime factor uncertainty. The S-characteristic has been introduced on the model of the L-characteristic by replacing the standard metric by a degenerate. Other than that the procedure remains very much the same.

In the present paper we take the natural step of generalizing the construction to the case of an indefinite metric, a metric induced by the matrix

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

∗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

It turns out that this metric is the right one for the study of the class of bounded real functions, namely the analytic functions in the open right half plane which have norm bound of less than one.

Our object in this paper is to study the class of bounded real functions via their normalized coprime factorizations, the normalization being with respect to the indefinite metric. Once we have the normalized coprime factors we can, using the Youla-Kucera parametrization, proceed with the construction of the B-characteristic. The B-characteristic can be used in the analysis of a special balancing which uses the positive definite solutions to the bounded real Riccati equations rather than the standard Lyapunov equations used by Moore [1981] in the case of stable transfer functions or the standard Riccati equations used by Jonckheere and Silverman [1983] for LQG balancing.

However our main object will be to focus on the application of the B-characteristic to the problem of parametrizing all suboptimal solutions to the Nehari complementation problem. This problem, solved originally by Nehari [1957], characterized the class of bounded Hankel operators. It has long been known that some classical interpolation problems, connected with the work of Schur, Nevanlinna and Pick, Fejer and Caratheodory etc, could be cast in the framework of Nehari’s theorem. In a seemingly independent line of research, starting with Beurling’s theorem, Beurling [1949] and Lax [1959], and the work of M.S. Livsic [1954] on characteristic functions, a deep connection was made between problems of interpolation and lifting of operators. This seems to have been initialized in the work of Sz.-Nagy and Koranyi [1958]. However the big advance was made by Sarason [1967] who proved the first version of the commutant lifting theorem. This was quickly generalized by Sz.-Nagy and Foias [1970] who proved the version of the commutant lifting theorem that is used widely today. Almost simultaneously with the availability of this theorem came the observation, due to Page [1970], that Nehari’s theorem could be derived from the commutant lifting theorem.

Parallel to this, operator theoretic oriented, development there was a direct study of Hankel operators and their relevance to various interpolation and approximation problems. This development initialized by a series of seminal papers by Adamjan, Arov and Krein [1968a,1968b,1971,1978], is currently being referred to as AAK theory. Though it seems a bit less general, AAK theory can be easily used to derive the most useful versions of the commutant lifting theorem. This is based on a very close connection between certain classes of Hankel operators and the model operators based on compressions of shifts, see Sz.-Nagy and Foias [1970], Rota [1960], Fuhrmann [1981] and Nikolskii [1985]. In the early eighties, in the wake of the influential work of Zames [1981] and its change of emphasis in control problems, came the recognition of the great relevance of AAK theory to control and this triggered a vast amount of research. This came to be known as H^{∞}-control. Probably the most influential paper in this connection is Glover [1984] which brought AAK theory into the realm of standard state space theory. While, for the finite dimensional situation, it became possible to develop AAK theory almost solely from the matrix theoretic point of view, this tendency has the disadvantage that it loses many of the geometric insights that the operator theoretic approach has to offer.

Mostly in the model matching problem, see Francis [1987], and in other problems that can be reduced to it, the role of Nehari complementation is central. Over the years several distinct methods have been put forward to solve the Nehari extension problem. These include the one step extension methods of Adamjan, Arov and Krein [1978], Young [1988], the method of solving a set of equations that have are induced by the Hankel operator and its adjoint, a method that originated with Krein and Melik-Adamyan [1984] see also Dym [1989], methods based on geometric considerations and J-spectral factorizations as in Ball and Ran [1983] and Francis [1987], state space methods based on balanced realizations and the solution of pairs of Lyapunov equations as in Glover [1984]. Another method used by Adamjan, Arov and Krein [1971] is based on Schmidt pair analysis. This has been put, for the rational case, into algebraic form using polynomial based methods in Fuhrmann [1991,1993a].

It is a proof of the richness of this circle of ideas that so many seemingly different methods converge on the same problem. However not much attention has been paid to elucidating the connections between the various methods. In this paper we will propose yet another approach to this problem, based on the analysis and use of the B-characteristic and embedding results. We will not stop with that but study the connection of this method with the Krein and Melik-Adamyan method as well as the method based on J-spectral factorization. The main advantage of the approach presented in this paper is not only its relative simplicity but the way it unifies several other approaches to Nehari extensions. For a problem area to be fully understood one needs to be able to view the results from several perspectives. It is strongly felt by the author that the perspective presented in this paper, with its emphasis on coprime factorizations, model operators, intertwining maps and Hankel operators, all leading directly to state space formulas based on the shift realization, is a particularly advantegeous perspective. Moreover, the material presented in section 3 opens up interesting possibilities of extending AAK theory to other contexts.

Some of the work presented in this paper was done during a visit to the Department of Mathematics at the University of Kaiserslautern. The hospitality of the department and in particular of my host D. Prätzel-Wolters are gratefully acknowledged. I am also indebted to the DFG for its support. Finally I want to thank Raimund Ober for the numerous discussions during the work on this paper.

2 Bounded real functions

In this section we quote several results on the special normalized coprime factorizations associated with bounded real functions that lead quite naturally to the construction of the bounded real characteristic. All the material in this section is based on Fuhrmann and Ober [1993b] and Ober and Fuhrmann [1993]. We arrange the results in a series of lemmas. For proofs we refer to the above mentioned papers.

Definition 2.1 1. A rational transfer function G is called bounded real if it is analytic and strictly contractive in the closed right half plane i.e.

I − G(s)^{∗}G(s) > 0

We assume this inequality holds also at infinity.

2. Let

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

A right coprime factorization G = NM^{−1} of G is called a J_{B} normalized right coprime factorization, or J_{B}-NRCF, of G if

( M^{∗} N^{∗} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M \\ N \end{matrix} ) = M^{∗}M − N^{∗}N = I.

Similarly a left coprime factorization G = \overline{M}^{−1}\overline{N} of the transfer function G is called a J_{B}-NLCF of G if

\overline{M}\overline{M}^{∗} − \overline{N}\overline{N}^{∗} = I.

We have the following useful lemmas.

Lemma 2.1 Let G = NM^{−1} be a J_{B}-NRCF and G = \overline{M}^{−1}\overline{N} a J_{B}-NLCF of the transfer function G. Then

1.  ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) .  (1)

2.  ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} )  (2)

3.  ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & −\overline{N}^{∗} \\ −N & \overline{M}^{∗} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )  (3)

Definition 2.2 Let G = NM^{−1} be a J_{B}-NRCF and G = \overline{M}^{−1}\overline{N} a J_{B}-NLCF of the transfer function G and let K = UV^{−1} be a RCF and K = \overline{V}^{−1}\overline{U} a LCF of the transfer function K. If

( \begin{matrix}\overline{V} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & U \\ N & V \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

holds, then the coprime factorizations are said to satisfy the J_{B}-doubly coprime factorization of G and K.

Of course doubly coprime factorizations are related to the Youla-Kucera parametrization of stabilizing controllers, see Francis [1987] for the details. The next lemma relates coprime factorizations of the plant and controller to doubly coprime factorizations.

Lemma 2.2 Let G = NM^{−1} be a J_{B} − NRCF and G = \overline{M}^{−1}\overline{N} a J_{B} − NLCF of the transfer function G and let K = UV^{−1} be a RCF and K = \overline{V}^{−1}\overline{U} a LCF of the function K. These coprime factorizations satisfy the J_{B}-doubly coprime factorization if and only if

\overline{V} M − \overline{U}N = I

and

\overline{M}V − \overline{N}U = I.

Lemma 2.3 Let G = NM^{−1} be a J_{B} − NRCF and G = \overline{M}^{−1}\overline{N} a J_{B} − NLCF of the transfer function G.

1. Consider the Bezout equation \overline{M}V − \overline{N}U = I. Then,

(a) there exists a unique solution (U_{B}, V_{B}) ∈ H_{+}^{∞} to the Bezout equation such that

R_{B}^{∗} := M^{∗}U_{B} − N^{∗}V_{B} ∈ H_{−}^{∞},

and is strictly proper.

(b) let (U, V) ∈ H_{+}^{∞} be an arbitrary solution to the Bezout equation then R_{B}^{∗} is the strictly proper antistable part of M^{∗}U − N^{∗}V.

(c) the Hankel operator

H_{M^{∗}U−N^{∗}V}

is independent of the solution to the Bezout equation.

2. Consider the Bezout equation \overline{V} M − \overline{U}N = I. Then,

(a) there exists a unique solution (\overline{U}_{B}, \overline{V}_{B}) ∈ H_{+}^{∞} to the Bezout equation such that

\overline{R}_{B}^{∗} := \overline{U}_{B}\overline{M}^{∗} − \overline{V}_{B}\overline{N}^{∗} ∈ H_{−}^{∞},

and is strictly proper.

(b) let (\overline{U}, \overline{V}) ∈ H_{+}^{∞} be an arbitrary solution to the Bezout equation then \overline{R}_{B}^{∗} is the strictly proper antistable part of \overline{U}\overline{M}^{∗} − \overline{V} \overline{N}^{∗}.

(c) the Hankel operator

H_{\overline{U}\overline{M}^{∗}−\overline{V} \overline{N}^{∗}}

is independent of the solution to the Bezout equation.

Proposition 2.1 Let G = NM^{−1} be a J_{B} − NRCF and G = \overline{M}^{−1}\overline{N} a J_{B} − NLCF of the transfer function G and let K_{1} = U_{1}V_{1}^{−1} be a RCF of the transfer function K and K_{2} = \overline{V} _{2}^{−1}\overline{U}_{2} a LCF of the transfer function K_{2} such that

\overline{M}V_{1} − \overline{N}U_{1} = I

and

\overline{V}_{2}M − \overline{U}_{2}N = I.

Then the following are equivalent:

1.  K_{1} = K_{2}.
2.  −N^{∗}V_{1} + M^{∗}U_{1} = −\overline{V}_{2}\overline{N}^{∗} + \overline{U}_{2}\overline{M}^{∗}
3.  ( \begin{matrix}−\overline{U}_{2} & \overline{V}_{2} \\ \overline{M} & −\overline{N} \end{matrix} ) ( \begin{matrix}N & V_{1} \\ M & U_{1} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

Proof: Using Lemma 2.1 we have that

( \begin{matrix}N & \overline{M}^{∗} \\ M & \overline{N}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}N^{∗} & M^{∗} \\ \overline{M} & \overline{N} \end{matrix} ) ( \begin{matrix}−I & 0 \\ 0 & I \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) .

The proposition follows from the following equality.

( \begin{matrix}I & \overline{V}_{2}U_{1} − \overline{U}_{2}V_{1} \\ 0 & I \end{matrix} ) = ( \begin{matrix}−\overline{U}_{2} & \overline{V}_{2} \\ \overline{M} & −\overline{N} \end{matrix} ) ( \begin{matrix}N & V_{1} \\ M & U_{1} \end{matrix} )

= ( \begin{matrix}−\overline{U}_{2} & \overline{V}_{2} \\ \overline{M} & −\overline{N} \end{matrix} ) ( \begin{matrix}N & \overline{M}^{∗} \\ M & \overline{N}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}N^{∗} & M^{∗} \\ \overline{M} & \overline{N} \end{matrix} ) ( \begin{matrix}−I & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}N & V_{1} \\ M & U_{1} \end{matrix} )

= ( \begin{matrix}I & \overline{V}_{2}\overline{N}^{∗} − \overline{U}_{2}\overline{M}^{∗} \\ 0 & I \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}I & −N^{∗}V_{1} + M^{∗}U_{1} \\ 0 & −I \end{matrix} )

= ( \begin{matrix}I & [−N^{∗}V_{1} + M^{∗}U_{1}] − [−\overline{V}_{2}\overline{N}^{∗} + \overline{U}_{2}\overline{M}^{∗}] \\ 0 & I \end{matrix} ) .

Corollary 2.1 We have

1.  R_{B}^{∗} = \overline{R}_{B}^{∗}  (4)
2.  U_{B}V_{B}^{−1} = \overline{V} _{B}^{−1}\overline{U}_{B}  (5)
3.  ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )  (6)

4.  ( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ) ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )  (7)

Proof: From the proof of the proposition we have that

( \begin{matrix}I & \overline{V}_{B}U_{B} − \overline{U}_{B}V_{B} \\ 0 & I \end{matrix} ) = ( \begin{matrix}I & [−N^{∗}V_{1} + M^{∗}U_{1}] − [−\overline{V}_{2}\overline{N}^{∗} + \overline{U}_{2}\overline{M}^{∗}] \\ 0 & I \end{matrix} )

= ( \begin{matrix}I & R_{B}^{∗} − \overline{R}_{B}^{∗} \\ 0 & I \end{matrix} )

which implies that

\overline{V}_{B}U_{B} − \overline{U}_{B}V_{B} = R_{B}^{∗} − \overline{R}_{B}^{∗}.

Since the right hand side is in H_{−}^{∞} and is strictly proper and the left hand side is in H_{+}^{∞}, we have that both sides are zero which implies the result.

We will refer to R_{B} = \overline{R}_{B} as the bounded real characteristic of G, or equivalently, as the B-characteristic of G.

We note that equation (7) can be written also as

( \begin{matrix}M \\ N \end{matrix} ) ( \overline{V}_{B} −\overline{U}_{B} ) + ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) ( −\overline{N} \overline{M} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) (8)

Lemma 2.4 Let

R_{B}^{∗} = ( M^{∗} N^{∗} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} )

and

\overline{R}_{B}^{∗} = ( \overline{V}_{B} \overline{U}_{B} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} ) .

Then

1.  ( \overline{V}_{B} \overline{U}_{B} ) = R_{B}^{∗} ( \overline{N} \overline{M} ) + ( M^{∗} N^{∗} )  (9)

and

2.  ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) = ( \begin{matrix}M \\ N \end{matrix} ) \overline{R}_{B}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )  (10)

Proof:

1. We multiply the identity, which is derived from (3),

( \begin{matrix}M & −\overline{N}^{∗} \\ −N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

on the right by ( \begin{matrix}U_{B} \\ −V_{B} \end{matrix} ), to get

\begin{aligned}( \begin{matrix}U_{B} \\ −V_{B} \end{matrix} ) & = & ( \begin{matrix}M & −\overline{N}^{∗} \\ −N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}U_{B} \\ −V_{B} \end{matrix} ) = ( \begin{matrix}M & −\overline{N}^{∗} \\ −N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}R_{B}^{∗} \\ −I \end{matrix} ) \\ & = & ( \begin{matrix}M \\ −N \end{matrix} ) R_{B}^{∗} − ( \begin{matrix}−\overline{N}^{∗} \\ \overline{M} & ∗ \end{matrix} ) \end{aligned}

or

( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) = ( \begin{matrix}M \\ N \end{matrix} ) R_{B}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )

2. Similarly, multiplying the same identity on the left by ( \overline{V}_{B} \overline{U}_{B} ), we get

( \overline{V}_{B} \overline{U}_{B} ) = ( \overline{V}_{B} \overline{U}_{B} ) ( \begin{matrix}M & −\overline{N}^{∗} \\ −N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} )
= ( I R_{B}^{∗} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} )
= ( M^{∗} N^{∗} ) + R_{B}^{∗} ( \overline{N} \overline{M} )

Lemma 2.5 Let G be a bounded real, proper, rational function and let

G = NM^{−1} = \overline{M}^{−1}\overline{N} be its normalized J_{B} coprime factorizations. Then

1.  \{ \begin{matrix}MM^{∗} − \overline{N}^{∗}\overline{N} = I \\ MN^{∗} = \overline{N}^{∗}\overline{M} \\ NM^{∗} = \overline{M}^{∗}\overline{N} \\ \overline{M}^{∗}\overline{M} − NN^{∗} = I. \end{matrix}  (11)

2. The function M^{−1}\overline{N}^{∗} is in the unit ball of L^{∞}.

3. We have the following equality − M^{−1}\overline{N}^{∗} = R_{B}^{∗} − M^{−1}U_{B} (12)

4. The singular values of H_{R_{B}^{∗}} are of absolute value less than 1.

Proof:

1. Follows from the identity, which is equivalent to (3),

( \begin{matrix}M & −\overline{N}^{∗} \\ −N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

2. From the equality MM^{∗} − \overline{N}^{∗}\overline{N} = I we get

I − (M^{−1}\overline{N}^{∗})(\overline{N}M^{−∗}) = M^{−1}M^{−∗} > 0. So M^{−1}\overline{N}^{∗} is contractive on the imaginary axis, i.e. it is in the unit ball of L^{∞}.

3. From the equality

( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) = ( \begin{matrix}M \\ N \end{matrix} ) \overline{R}_{B}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )

we get

M^{−1}U_{B} = R_{B}^{∗} + M^{−1}\overline{N}^{∗}

or (12) follows.

4. Since M^{−1}U_{B} is stable we have H_{R_{B}^{∗}} = −H_{M^{−1}\overline{N}^{∗}} and so

μ_{1} = ∥H_{R_{B}^{∗}}∥ = ∥H_{M^{−1}\overline{N}^{∗}}∥ ≤ ∥M^{−1}\overline{N}^{∗}∥_{∞} < 1.

This, of course, implies that all the singular values of H_{R_{B}^{∗}} are less than 1.

We want to point out that equation (12) i.e.

−M^{−1}\overline{N}^{∗} = R_{B}^{∗} − M^{−1}U_{B}

shows that the function R_{B}^{∗} is the strictly proper, antistable part of the contractive L^{∞} function −M^{−1}\overline{N}^{∗}. In the language of Nehari’s theorem this shows that −M^{−1}\overline{N}^{∗} is a suboptimal Nehari extension of R_{B}^{∗}. In fact this simple equality points out the way to yet another aapproach to the Nehari extension problem. We will return to this subject in section 5.

3 Hankel operators

In this section we study several Hankel operators associated with the normalized coprime factorizations of a bounded real functions. Many of the results are based on a similar analysis carried out in Fuhrmann and Ober [1993a].

As usual, given a matrix function Φ ∈ L^{∞} we define the Hankel operator H_{Φ} : H_{+}^{2} ⟶ H_{−}^{2} by

H_{Φ}f = P_{−}Φf.

Similarly, the involuted Hankel operator Ĥ_{Φ} : H_{−}^{2} ⟶ H_{+}^{2} is defined by

Ĥ_{Φ}h = P_{+}Φh.

Clearly we have H_{Φ}^{∗} = Ĥ_{Φ^{∗}}.

Let now R_{B}^{∗} be the B-characteristic of the bounded real function G. As R_{B}^{∗} is rational and in H_{−}^{∞} it has coprime Douglas, Shapiro and Shields [1971], or DSS, factorizations of the form

R_{B}^{∗} = Φ_{K}S_{K}^{∗} = S_{I}^{∗}Φ_{I} (13) with S_{K} and S_{I} inner functions in H_{+}^{∞}. For more on these factorizations we refer to Fuhrmann [1975,1981] and Fuhrmann and Ober [1993b]. We know that these factorizations imply

\{ \begin{matrix}KerH_{R^{∗}} & = & S_{K}H_{+}^{2} \\ ImH_{R^{∗}} & = & \{S_{I}^{∗}H_{−}^{2}\}^{⊥} = H_{−}^{2} ⊖ S_{I}^{∗}H_{−}^{2} \end{matrix}

and the induced map from {S_{K}H_{+}^{2}}^{⊥} to {S_{I}^{∗}H_{−}^{2}}^{⊥} is invertible. Moreover, by the rationality of R_{B}^{∗}, these are finite dimensional spaces.

Given an antistable function A^{∗} we will say that an inner function S is a minimal left inner function of A^{∗} if the following conditions are satisfied

1. SA^{∗} ∈ H_{+}^{∞}

2. For any inner function T for which T A^{∗} ∈ H_{+}^{∞} we have T = QS.

A minimal right inner function is defined similarly.

Lemma 3.1 Let G be a bounded real function with J_{B} normalized coprime factorizations

G = NM^{−1} = \overline{M}^{−1}\overline{N} (14)

and let R_{B}^{∗} be its BR-characteristic. Assume (13) are the DSS factorizations of R^{∗}. Then

1. The minimal right inner function of ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) is S_{K}.

2. The minimal left inner function of ( N^{∗} M^{∗} ) is S_{I}.

Proof:

1. Since KerH_{R^{∗}} = S_{K}H_{+}^{2} we have

P_{−}R_{B}^{∗}S_{K}H_{+}^{2} = P_{−} ( \overline{V}_{B} −\overline{U}_{B} ) ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) S_{K}H_{+}^{2} = 0.

However it is clear that also

P_{−} ( −\overline{N} \overline{M} ) ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) S_{K}H_{+}^{2} = 0.

Since ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) is invertible in H_{+}^{∞} we have, for all f ∈ H_{+}^{2},

\begin{aligned}0 & = & P_{−} ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) S_{K}f \\ & = & P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) S_{K}f \end{aligned}

This implies ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) S_{K} ∈ H_{+}^{∞}. We set

( \begin{matrix}J_{1} \\ J_{2} \end{matrix} ) = ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) S_{K}. (15)

Let now Q be any inner function for which ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) Q ∈ H_{+}^{∞}. This implies that for an arbitrary f ∈ H_{+}^{2} we have P_{−} ( \overline{V}_{B} −\overline{U}_{B} ) ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) Qf = 0. But we also have for an arbitrary f ∈ H_{+}^{2} that P_{−} ( −\overline{N} \overline{M} ) ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) Qf = P_{−}Qf = 0. It follows as before that H_{R^{∗}}f = P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) Qf = 0, i.e. Qf ∈ KerH_{R^{∗}} = S_{K}H_{+}^{2}. Thus we have the inclusion QH_{+}^{2} ⊂ S_{K}H_{+}^{2}. This implies the factorization Q = S_{K}T for some inner function T.

2. The proof is similar and omitted. We set ( K_{1} K_{2} ) = S_{I} ( N^{∗} M^{∗} ) (16)

Corollary 3.1 Let S_{K} be the minimal right inner function of ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ), then

KerH( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) = S_{K}H_{+}^{2}. (17)

Proof: Clearly we have S_{K}H_{+}^{2} ⊂ KerH( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ). To see the converse inclusion we note that, by Beurling’s theorem, KerH( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) = QH_{+}^{2} for some inner function Q. Therefore,

for an arbitrary f ∈ H_{+}^{2}, we have P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) Qf = 0. This implies that ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) Q ∈ H_{+}^{∞}. So, by the minimality of S_{K} we have the factorization Q = S_{K}T, i.e.

KerH( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) = QH_{+}^{2} ⊂ S_{K}H_{+}^{2}.

The two inclusions, taken together, imply equality (17).

Given the p × m bounded real function G with the J_{B}-normalized coprime factorizations (14), and with J_{i} defined by (15), we define the (p + m) × (p + m) function Ω_{K} by

Ω_{K} = ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) (18)

Next we consider the H_{+}^{2} space of C^{p+m} valued functions. On this space we define a new, indefinite, inner product by letting

[( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) , ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )] = (J_{B} ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) , ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )) = (f_{1}, g_{1}) − (f_{2}, g_{2}) (19)

We denote this space, supressing in the notation the dependence on p and m, by H_{+}^{2,J}. Thus H_{+}^{2,J} is a Krein space, see Bognar [1974] for the general theory of these spaces.

In H_{+}^{2,J} we define a map, based on the matrix function Ω_{K}. Thus we set, for f ∈ H_{+}^{2,J},

P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}f = Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}f (20)

Then we have

Proposition 3.1 Let P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}} be defined by (20).

1. The following identities hold.

\{ \begin{matrix}Ω_{K}^{∗}J_{B}Ω_{K} & = & J_{B} \\ Ω_{K}^{∗}J_{B}Ω_{K}J_{B} & = & I \\ Ω_{K}J_{B}Ω_{K}^{∗}J_{B} & = & I \\ Ω_{K}J_{B}Ω_{K}^{∗} & = & J_{B} \end{matrix} (21)

2. The map P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}} is a projection map in H_{+}^{2,J}.

3. We have

KerP_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}} = Ω_{K}H_{+}^{2,J} (22)

4. We have

{Ω_{K}H_{+}^{2,J}}^{[⊥]} = {f ∈ H_{+}^{2,J}|Ω_{K}^{∗}J_{B}f ∈ H_{−}^{2}} (23)

5. We have

ImP_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}} = {Ω_{K}H_{+}^{2,J}}^{[⊥]} (24)

6. We have

H_{+}^{2,J} = {Ω_{K}H_{+}^{2,J}}^{[⊥]}[⊕]Ω_{K}H_{+}^{2,J} (25)

Proof:

1. These identities follow by direct computation.

2. We use the identity Ω_{K}^{∗}J_{B}Ω_{K} = J_{B} to compute

(P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}})^{2}f = Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}f
= Ω_{K}J_{B}P_{−}J_{B}^{2}P_{−}Ω_{K}^{∗}J_{B}f
= Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}f = P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}f

3. Let f ∈ Ω_{K}H_{+}^{2,J}, with f = Ω_{K}g, then

P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}f = P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}Ω_{K}g
= Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}Ω_{K}g
= Ω_{K}J_{B}P_{−}J_{B}g = 0

as J_{B}g ∈ H_{+}^{2}. This shows that Ω_{K}H_{+}^{2,J} ⊂ KerP_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}.

Conversely, assume f ∈ KerP_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}. Then

0 = P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}f = Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}f This implies g = Ω_{K}^{∗}J_{B}f ∈ H_{+}^{2}. But then Ω_{K}J_{B}g = Ω_{K}J_{B}Ω_{K}^{∗}J_{B}f = f, i.e. f ∈ Ω_{K}H_{+}^{2,J}.

4. Assume f ∈ {Ω_{K}H_{+}^{2,J}}^{[⊥]} i.e. [f, Ω_{K}g] = 0 for all g ∈ H_{+}^{2,J}. Then

0 = [f, Ω_{K}g] = (J_{B}f, Ω_{K}g) = (Ω_{K}^{∗}J_{B}f, g) i.e. Ω_{K}^{∗}J_{B}f ∈ H_{−}^{2}.

5. We have f ∈ ImP_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}} if and only if f = P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}f = Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}f. However this equality holds if and only if Ω_{K}^{∗}J_{B}f ∈ H_{−}^{2}.

6. Follows from the characterization of the Kernel and Image of the projection P_{{Ω_{K}H_{+}^{2,J} }^{[⊥]}}.

In a similar way, given the p × m bounded real function G with the J_{B}-normalized coprime factorizations (14). Let us define

( K_{1} K_{2} ) = S_{I} ( N^{∗} M^{∗} ) (26)

We define the (p + m) × (p + m) function Ω_{I} by

Ω_{I} = ( \begin{matrix}\overline{M} & \overline{N} \\ K_{1} & K_{2} \end{matrix} ) (27)

Also, we consider the H_{−}^{2} space of C^{p+m} valued functions. On this space we define a new, indefinite, inner product by letting

[( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) , ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )] = (J_{B} ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) , ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )) = (f_{1}, g_{1}) − (f_{2}, g_{2}) (28)

We denote this space, supressing in the notation the dependence on p and m, by H_{−}^{2,J}. Thus H_{−}^{2,J} is also a Krein space.

In H_{−}^{2,J} we define a map, based on the matrix function Ω_{I}. Thus we set, for h ∈ H_{−}^{2,J},

P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}}h = Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B}h (29)

Now we can prove the counterpart to Proposition 3.1.

Proposition 3.2 Let P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}} be defined by (29).

1. The following identities hold.

\{ \begin{matrix}Ω_{I}J_{B}Ω_{I}^{∗} & = & J_{B} \\ Ω_{I}^{∗}J_{B}Ω_{I}J_{B} & = & I \\ Ω_{I}J_{B}Ω_{I}^{∗}J_{B} & = & I \\ Ω_{I}^{∗}J_{B}Ω_{I} & = & J_{B} \end{matrix} (30)

2. The map P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}} is a projection map in H_{−}^{2,J}.

3. We have

KerP_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}} = Ω_{I}^{∗}H_{−}^{2,J} (31)

4. We have

{Ω_{I}^{∗}H_{−}^{2,J}}^{[⊥]} = {h ∈ H_{−}^{2,J}|Ω_{I}J_{B}h ∈ H_{+}^{2}} (32)

5. We have

ImP_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}} = {Ω_{I}^{∗}H_{−}^{2,J}}^{[⊥]} (33)

6. We have

H_{+}^{2,J} = {Ω_{I}^{∗}H_{−}^{2,J}}^{[⊥]}[⊕]Ω_{I}^{∗}H_{−}^{2,J} (34)

Proof:

1. The identities follow by direct computation.

2. We use the identity Ω_{I}J_{B}Ω_{I}^{∗} = J_{B} to compute

(P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}})^{2}f = Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B}Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B}f
= Ω_{I}^{∗}J_{B}P_{+}J_{B}^{2}P_{+}Ω_{I}J_{B}f
= Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B}f = P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}}f

3. Let f ∈ Ω_{I}^{∗}H_{−}^{2,J}, with f = Ω_{I}^{∗}g, then

P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}}f = P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}}Ω_{I}^{∗}g
= Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B}Ω_{I}^{∗}g
= Ω_{I}^{∗}J_{B}P_{+}J_{B}g = 0

as J_{B}g ∈ H_{−}^{2}. This shows that Ω_{I}^{∗}H_{−}^{2,J} ⊂ KerP_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}}.

4. Assume f ∈ {Ω_{I}^{∗}H_{−}^{2,J}}^{[⊥]} i.e. [f, Ω_{I}^{∗}g] = 0 for all g ∈ H_{−}^{2,J}. Then

0 = [f, Ω_{I}^{∗}g] = (J_{B}f, Ω_{I}^{∗}g) = (Ω_{I}J_{B}f, g) i.e. Ω_{I}J_{B}f ∈ H_{+}^{2}.

5. We have f ∈ ImP_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}} if and only if f = P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}}f = Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B}f. However this equality holds if and only if Ω_{I}J_{B}f ∈ H_{+}^{2}.

6. Follows from the characterization of the Kernel and Image of the projection P_{{Ω_{I}^{∗}H_{−}^{2,J} }^{[⊥]}}.

Given a linear transformation T in a Krein space we define its adjoint with respect to the indefinite metric as the unique transformation T^{♯} satisfying, for all x, y,

[T x, y] = [x, T^{♯}y] (35)

Proposition 3.3 For

Ω_{K} = ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) (36)

we define the map Z_{K} : {S_{K}H_{+}^{2}}^{⊥} ⟶ {Ω_{K}H_{+}^{2,J}}^{[⊥]} by

Z_{K}f = P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f for f ∈ \{S_{K}H_{+}^{2}\}^{⊥} (37)

and the map Y_{K} : {Ω_{K}H_{+}^{2,J}}^{[⊥]} ⟶ {S_{K}H_{+}^{2}}^{⊥} by

Y_{K} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = −P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) for ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) ∈ \{S_{K}H_{+}^{2}\}^{⊥} (38)

Then 1. The map Z is invertible and Z_{K}^{−1} = Y_{K}.

2. The indefinite adjoint of Z_{K} is given by

Z_{K}^{♯} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = P_{\{S_{K}H_{+}^{2}\}^{⊥}}P_{+} ( U_{B}^{∗} V_{B}^{∗} ) J_{B} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) for ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) ∈ \{Ω_{K}H_{+}^{2,J}\}^{[⊥]}(39)

3. The indefinite adjoint of Y_{K} is given by

Y_{K}^{♯}f = −J_{B}P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}}P_{+} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f for f ∈ \{S_{K}H_{+}^{2}\}^{⊥} (40)

Proof: 1. For f ∈ {S_{K}H_{+}^{2}}^{⊥} we have

Y_{K}Z_{K}f = −P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= −P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( −\overline{N} \overline{M} ) ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) J_{B}P_{−} ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= S_{K}P_{−}S_{K}^{∗} ( 0 S_{K} ) J_{B}P_{−} ( \begin{matrix}R_{B}^{∗} \\ −S_{K}^{∗} \end{matrix} ) f
= S_{K}P_{−} ( 0 −I ) P_{−} ( \begin{matrix}R_{B}^{∗} \\ −S_{K}^{∗} \end{matrix} ) f
= S_{K}P_{−}S_{K}^{∗}f = P_{{S_{K}H_{+}^{2}}^{⊥}}f = f

Similarly, for ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) \{Ω_{K}H_{+}^{2,J}\}^{[⊥]}, and using equality (8) we have

Z_{K}Y_{K} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) (−1)P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) S_{K}P_{−}S_{K}^{∗} ( −\overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= Ω_{K}J_{B}P_{−} ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) ( \begin{matrix}U_{B} \\ −V_{B} \end{matrix} ) S_{K}P_{−}S_{K}^{∗} ( −\overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= Ω_{K}J_{B}P_{−} ( \begin{matrix}R_{B}^{∗} \\ −S_{K}^{∗} \end{matrix} ) S_{K}P_{−}S_{K}^{∗} ( −\overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= Ω_{K}J_{B}P_{−}J_{B} ( \begin{matrix}R_{B}^{∗} \\ S_{K}^{∗} \end{matrix} ) ( −\overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}Ω_{K} ( \begin{matrix}R_{B}^{∗} \\ S_{K}^{∗} \end{matrix} ) ( −\overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} [( \begin{matrix}M \\ N \end{matrix} ) R_{B}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )] ( −\overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) ( −\overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}} \{( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) − ( \begin{matrix}M \\ N \end{matrix} ) ( \overline{V} −\overline{U} )\} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )
= ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )

2. Let f ∈ \{S_{K}H_{+}^{2}\}^{⊥}, ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) ∈ \{Ω_{K}H_{+}^{2,J}\}^{[⊥]}. Then

[Zf, ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} )] = (J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f, ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ))
= (f, P_{+} ( U_{B}^{∗} V_{B}^{∗} ) J_{B}Ω_{K}P_{−}J_{B}Ω_{K}^{∗}J_{B} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ))
= (f, P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}}P_{+} ( U_{B}^{∗} V_{B}^{∗} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ))

Thus (39) follows.

3. Let ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) ∈ \{Ω_{K}H_{+}^{2,J}\}^{[⊥]}, f ∈ \{S_{K}H_{+}^{2}\}^{⊥}. Then

(Y_{K} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) , f) = −(P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) , f)
= −(( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) , P_{+} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f)
= −(( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) , P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}}P_{+} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f)
= −[( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) , J_{B}P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}}P_{+} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f]

which proves (40).

Proposition 3.4 For

Ω_{I} = ( \begin{matrix}\overline{M} & \overline{N} \\ K_{1} & K_{2} \end{matrix} ) (41)

we define the map Z_{I} : {S_{I}^{∗}H_{−}^{2}}^{⊥} ⟶ {Ω_{I}^{∗}H_{−}^{2,J}}^{[⊥]} by

Z_{I}h = P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) h for h ∈ \{S_{I}^{∗}H_{−}^{2}\}^{⊥} (42)

and the map Y_{I} : {Ω_{I}^{∗}H_{−}^{2,J}}^{[⊥]} ⟶ {S_{I}^{∗}H_{−}^{2}}^{⊥} by

Y_{I} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) = −P_{\{S_{I}^{∗}H_{+}^{2}\}^{⊥}} ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) for ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) ∈ \{Ω_{I}^{∗}H_{−}^{2,J}\}^{[⊥]} (43)

Then 1. The maps Z_{I} and Y_{I} are invertible and Z_{I}^{−1} = Y_{I}.

2. The indefinite adjoint of Z_{I} is given by

Z_{I}^{♯} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) = P_{\{S_{I}^{∗}H_{−}^{2}\}^{⊥}}P_{−} ( \overline{U} \overline{V} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) for ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) ∈ \{Ω_{I}^{∗}H_{−}^{2,J}\}^{[⊥]}(44)

3. The indefinite adjoint of Y_{I} is given by

Y_{I}^{♯}h = −P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}}P_{−} ( \begin{matrix}N \\ M \end{matrix} ) h for h ∈ \{S_{I}^{∗}H_{−}^{2}\}^{⊥} (45)

Proof:

1. For h ∈ {S_{I}^{∗}H_{−}^{2}}^{⊥} we have

Y_{I}Z_{I} = −P_{\{S_{I}^{∗}H_{+}^{2}\}^{⊥}} ( N^{∗} M^{∗} ) J_{B}P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) h
= P_{\{S_{I}^{∗}H_{+}^{2}\}^{⊥}} ( −N^{∗} M^{∗} ) P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) h
= S_{I}^{∗}P_{+}S_{I} ( −N^{∗} M^{∗} ) ( \begin{matrix}\overline{M}^{∗} & K_{1}^{∗} \\ \overline{N}^{∗} & K_{2}^{∗} \end{matrix} ) J_{B}P_{+} ( \begin{matrix}\overline{M} & \overline{N} \\ K_{1} & K_{2} \end{matrix} ) J_{B} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) h
= S_{I}^{∗}P_{+}S_{I} ( 0 −S_{I}^{∗} ) P_{+} ( \begin{matrix}R_{B} \\ −S_{I} \end{matrix} ) h = h.

On the other hand, for ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) ∈ \{Ω_{I}^{∗}H_{−}^{2,J}\}^{[⊥]}, we compute, using equality (9) as well as the following equality

( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) ( −N^{∗} M^{∗} ) + ( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} ) ( V^{∗} −U^{∗} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

Z_{I}Y_{I} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) = −P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) P_{\{S_{I}^{∗}H_{+}^{2}\}^{⊥}} ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= −Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) = S_{I}^{∗}P_{+}S_{I} ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= −Ω_{I}^{∗}J_{B}P_{+} ( \begin{matrix}R \\ −S_{I} \end{matrix} ) ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= −Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B}Ω_{I}^{∗} ( \begin{matrix}R \\ S_{I} \end{matrix} ) ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= −P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} [( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} ) R_{B} + ( \begin{matrix}N \\ M \end{matrix} )] ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= −P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}\overline{U} & ∗ \\ \overline{V} & ∗ \end{matrix} ) ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}\overline{U} & ∗ \\ \overline{V} & ∗ \end{matrix} ) ( −N^{∗} M^{∗} ) ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} \{( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) − ( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} ) ( V^{∗} −U^{∗} )\} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )
= ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )

2. Let h ∈ \{S_{I}^{∗}H_{+}^{2}\}^{⊥} and ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) ∈ \{Ω_{I}^{∗}H_{−}^{2,J}\}^{[⊥]}. Then

[Z_{I}h, ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} )] = (J_{B}P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) h, ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ))
= (Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B} ( \begin{matrix}\overline{U} & {∗ \atop B} \\ \overline{V} & {∗ \atop B} \end{matrix} ) h, J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ))
= (h, ( \overline{U}_{B} \overline{V}_{B} ) J_{B}Ω_{I}^{∗}P_{+}J_{B}Ω_{I}J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ))
= (h, P_{\{S_{I}^{∗}H_{+}^{2}\}^{⊥}} ( \overline{U}_{B} \overline{V}_{B} ) J_{B}Ω_{I}^{∗}P_{+}J_{B}Ω_{I}J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ))

This proves (44).

3. Let h ∈ \{S_{I}^{∗}H_{+}^{2}\}^{⊥} and ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) ∈ \{Ω_{I}^{∗}H_{−}^{2,J}\}^{[⊥]}. Then

(Y_{I} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , h) = (−P_{\{S_{I}^{∗}H_{+}^{2}\}^{⊥}} ( N^{∗} M^{∗} ) J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , h)
= (( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , ( \begin{matrix}−N \\ M \end{matrix} ) h) = (P_{−} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , ( \begin{matrix}−N \\ M \end{matrix} ) , h)
= (( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , P_{−} ( \begin{matrix}−N \\ M \end{matrix} ) , h)
= (P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , P_{−} ( \begin{matrix}−N \\ M \end{matrix} ) , h)
= (Ω_{I}^{∗}J_{B}P_{+}Ω_{I}J_{B} ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , P_{−} ( \begin{matrix}−N \\ M \end{matrix} ) , h)
= (( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , J_{B}Ω_{I}^{∗}P_{+}J_{B}Ω_{I}J_{B}^{2}P_{−} ( \begin{matrix}−N \\ M \end{matrix} ) , h)
= [( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}}J_{B}P_{−} ( \begin{matrix}−N \\ M \end{matrix} ) , h]
= − [( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}}P_{−} ( \begin{matrix}N \\ M \end{matrix} ) , h]

This proves (45). In Proposition 3.3 we studied the matrix function Ω_{K} = ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) in great detail. The reason for this is that it provides the representation of the kernel of an important Hankel operator, namely the operator H_{( _{M^{∗} N^{∗}} )J_{B}} = H_{( _{M^{∗} −N^{∗}} )}. We make this more specific in the following. Proposition 3.5 1. With the operator Z_{K} defined by (37), we have

H_{R_{B}^{∗}} = H( _{M^{∗} N^{∗}} ) J_{B}Z_{K} (46)

2. We have

KerH( _{M^{∗} N^{∗}} ) J_{B} = Ω_{K}H_{+}^{2,J} (47)

Proof:

1. We compute

H( _{M^{∗} N^{∗}} ) J_{B}Z_{K}f = P_{−} ( M^{∗} N^{∗} ) J_{B}P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= P_{−} ( M^{∗} N^{∗} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= P_{−} ( M^{∗} N^{∗} ) J_{B} ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= P_{−} ( I 0 ) J_{B}P_{−} ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) ( \begin{matrix}U_{B} \\ −V_{B} \end{matrix} ) f
= P_{−} ( I 0 ) P_{−} ( \begin{matrix}R_{B}^{∗} \\ −S_{K}^{∗} \end{matrix} ) f
= H_{R_{B}^{∗}}f

2. Clearly we have the inclusion ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) H_{+}^{2,J} ⊂ KerH( _{M^{∗} N^{∗}} ) J_{B}.

Conversely, assume ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) ∈ KerH( _{M^{∗} N^{∗}} ) J_{B} and f ∈ \{Ω_{K}H_{+}^{2,J}\}^{[⊥]}. Then

( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = Zf for some f ∈ \{S_{K}H_{+}^{2}\}^{⊥}. Now H_{R_{B}^{∗}} = H( _{M^{∗} N^{∗}} ) J_{B}Z_{K} and the

invertibility of Z_{K} imply H_{R_{B}^{∗}}f = 0. But on {S_{K}H_{+}^{2}}^{⊥} the Hankel operator H_{R_{B}^{∗}} is

injective. So it follows that f = 0 and hence also ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = 0.

In the same way we can state the following. We omit the proof.

Proposition 3.6 With the operators Z_{I} and Y_{I} defined by (42) and (43) respectively, we have

1.  Ĥ_{R_{B}} = Ĥ( _{\overline{M} \overline{N}} ) J_{B}Z_{I}  (48)
2.  H( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} ) = Y_{I}^{♯}H_{R_{B}^{∗}}  (49)

3. We have

KerĤ( _{\overline{M} \overline{N}} ) J_{B} = Ω_{I}^{∗}H_{−}^{2,J} (50)

In analogy with the key diagram studied in great detail in Fuhrmann and Ober [1993] and in Fuhrmann [1993b] we have another equivalent diagram connecting the various Hankel and intertwining operators derived from the bounded real function G. Thus we have.

Proposition 3.7 The operator T : {Ω_{K}H_{+}^{2}}^{⊥} ⟶ {Ω_{I}^{∗}H_{−}^{2}}^{⊥}

( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) ↦ P_{−} ( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} ) (−\overline{N} \overline{M}) ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) (51)

is such that the following diagram commutes.

[figure: \{S_{K}H_{+}^{2}\}^{⊥}; \{Ω_{K}H_{+}^{2,J}\}^{⊥}; \{Ω_{I}^{∗}H_{−}^{2,J}\}^{⊥}; \{S_{I}^{∗}H_{−}^{2}\}^{⊥}; Z_{K}; Y_{I}^{♯}; H_{R_{B}}^{∗}; T; H_{(M^{∗} N^{∗})J_{B}}; H(_{\begin{matrix}\overline{M} & ^{∗} \\ \overline{N} & ^{∗} \end{matrix}} )]

Proof: In view of Propositions 3.5 and 3.6 it suffices to prove that

H( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} )Y_{K} = Y_{I}^{♯}Ĥ( _{\overline{M} \overline{N}} ) J_{B}. (52)

We begin by computing

H( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} )Y_{K} ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = (Y_{I}^{♯}H_{R_{B}^{∗}})Y_{K}

= Y_{I}^{♯}(H_{R_{B}^{∗}}Y_{K})
= Y_{I}^{♯}H( _{M^{∗} N^{∗}} ) J_{B}

Next we compute

Y_{I}^{♯}Ĥ( _{M^{∗} N^{∗}} ) J_{B} = P_{\{Ω_{I}^{∗}H_{−}^{2,J} \}^{[⊥]}}J_{B}P_{−} ( \begin{matrix}M \\ N \end{matrix} ) P_{−} ( M^{∗} N^{∗} ) J_{B}

As a corollary to the results in this section we have the following statement about the dimensions of the spaces with which we are dealing.

Corollary 3.2 If any of the spaces {S_{K}H_{+}^{2}}^{⊥}, {S_{I}^{∗}H_{−}^{2}}^{⊥}, {Ω_{K}H_{+}^{2,J}}^{⊥}, {Ω_{I}^{∗}H_{−}^{2,J}}^{⊥} is finite dimensional, so are all the other and

dim({S_{K}H_{+}^{2}}^{⊥}) = dim({S_{I}^{∗}H_{−}^{2}}^{⊥}) = dim({Ω_{K}H_{+}^{2,J}}^{⊥}) = dim({Ω_{I}^{∗}H_{−}^{2,J}}^{⊥}).

If the McMillan degree of G is n, then n is the dimension of these spaces.

Corollary 3.3 Let G be a rational, bounded real function of McMillan degree n. Then its B-characteristic R_{B}^{∗} has also McMillan degree n.

Proof: The McMillan degree of R_{B}^{∗} is equal to the dimension of any of the spaces {S_{K}H_{+}^{2}}^{⊥} or {S_{I}^{∗}H_{−}^{2}}^{⊥}. These are equal, by the previous corollary, to the dimensions of any of the spaces {Ω_{K}H_{+}^{2,J}}^{[⊥]} or {Ω_{I}^{∗}H_{−}^{2,J}}^{[⊥]}. These on the other hand are equal to the McMillan

degrees of ( M^{∗} N^{∗} ) or ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ). However, by the results of Fuhrmann and Ober [1993b],

these have the same McMillan degree as G.

Proposition 3.8 Let the operators Z_{K} and Y_{K} be defined by (37) and (38) respectively. Then the following identities are satisfied.

1.  Z_{K}^{♯}Z_{K} = H_{R_{B}^{∗}}^{∗}H_{R_{B}^{∗}} − I|_{{S_{K}H_{+}^{2}}^{⊥}}  (53)
2.  Y_{K}Y_{K}^{♯} = −I + H( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )^{∗}J_{B}H( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )|_{\{S_{K}H_{+}^{2}\}^{⊥}}  (54)

Proof:

1. We compute

Z_{K}^{♯}Z_{K} = P_{\{S_{K}H_{+}^{2}\}^{⊥}}P_{+} ( U_{B}^{∗} V_{B}^{∗} ) J_{B}P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= S_{K}P_{−}S_{K}^{∗}P_{+} ( U_{B}^{∗} V_{B}^{∗} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= S_{K}P_{−}S_{K}^{∗}P_{+} ( U_{B}^{∗} V_{B}^{∗} ) J_{B} ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) J_{B}P_{−} ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) J_{B} ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) f
= S_{K}P_{−}S_{K}^{∗}P_{+} ( R −S_{K} ) J_{B}P_{−} ( \begin{matrix}R^{∗} \\ −S^{∗} \end{matrix} ) f
= P_{{S_{K}H_{+}^{2}}^{⊥}}H_{R_{B}^{∗}}^{∗}H_{R_{B}^{∗}}f − S_{K}P_{−}S_{K}^{∗}P_{+}S_{K}P_{−}S_{K}^{∗}f

2. We compute

Y_{K}Y_{K}^{♯} = P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}P_{\{Ω_{K}H_{+}^{2,J} \}^{[⊥]}}P_{+} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f
= P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}(I − P_{−}) ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f
= P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

− P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

We compute each term separately.

P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}(I − P_{+})Ω_{K}^{∗}J_{B} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}Ω_{K}^{∗}J_{B} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

− P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}P_{+}Ω_{K}^{∗}J_{B} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f − P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( 0 S_{K} ) P_{+}Ω_{K}^{∗}J_{B} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= −f On the other hand

P_{\{S_{K}H_{+}^{2}\}^{⊥}} ( \overline{N} \overline{M} ) J_{B}Ω_{K}J_{B}P_{−}Ω_{K}^{∗}J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= S_{K}P_{−}S_{K}^{∗} ( \overline{N} \overline{M} ) J_{B} ( \begin{matrix}M & J_{1} \\ N & J_{2} \end{matrix} ) J_{B}P_{−} ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= S_{K}P_{−}S_{K}^{∗} ( 0 S_{K} ) J_{B}P_{−} ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= S_{K}P_{−} ( 0 −I ) P_{−} ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= S_{K}P_{−} ( 0 −I ) ( \begin{matrix}M^{∗} & N^{∗} \\ J_{1}^{∗} & J_{2}^{∗} \end{matrix} ) J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= −S_{K}P_{−}S_{K}^{∗} ( \overline{N} \overline{M} ) J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

= −S_{K}P_{−}S_{K}^{∗}P_{+} ( \overline{N} \overline{M} ) J_{B}P_{−} ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) f

Taken together, these computations imply (54). The next proposition is proved analogously. Proposition 3.9 Let the operators Z_{I} and Y_{I} be defined by (42) and (43) respectively. Then the following identities are satisfied.

1.  Z_{I}^{♯}Z_{I} = H_{R_{B}^{∗}}H_{R_{B}^{∗}}^{∗} − I|_{{S_{I}^{∗}H_{−}^{2}}^{⊥}}  (55)
2.  Y_{I}Y_{I}^{♯} = −I + H_{( _{N^{∗} M^{∗}} )}J_{B}Ĥ_{( \begin{matrix}N \\ M \end{matrix} )}|_{\{S_{I}^{∗}H_{−}^{2}\}^{⊥}}  (56)

The next theorem follows easily from the relations proved in the previous propositions. We omit the simple proof. This theorem has implications as far as balanced realizations are concerned, however we will not pursue this theme in the present paper.

Theorem 3.1 Let G be a rational, proper bounded real function and let R_{B} be its Bcharacteristic. Let 1 > μ_{1} ≥ ⋯ ≥ μ_{n} > 0 be the singular values of the Hankel operator H_{R_{B}^{∗}} and {f_{i}, h_{i}} the corresponding Schmidt pairs. We set

g_{i} = (1 − μ_{i}^{2})^{− \frac{1}{2}} Z_{K}f_{i} ∈ \{Ω_{K}H_{+}^{2,J}\}^{[⊥]} (57)

and

k_{i} = (1 − μ_{i}^{2})^{− \frac{1}{2}} Z_{I}h_{i} ∈ \{Ω_{I}^{∗}H_{−}^{2,J}\}^{[⊥]}. (58)

Then

1.  \{ \begin{matrix}Z_{K}f_{i} & = & (1 − μ_{i}^{2})^{\frac{1}{2}} g_{i} \\ Z_{K}^{♯}g_{i} & = & −(1 − μ_{i}^{2})^{\frac{1}{2}} f_{i} \end{matrix}  (59)

2.  \{ \begin{matrix}Y_{K}g_{i} & = & (1 − μ_{i}^{2})^{− \frac{1}{2}} f_{i} \\ Y_{K}^{♯}f_{i} & = & −(1 − μ_{i}^{2})^{− \frac{1}{2}} g_{i} \end{matrix}  (60)

3.  \{ \begin{matrix}Z_{I}h_{i} & = & (1 − μ_{i}^{2})^{\frac{1}{2}} k_{i} \\ Z_{I}^{♯}k_{i} & = & −(1 − μ_{i}^{2})^{\frac{1}{2}} h_{i} \end{matrix}  (61)

4.  \{ \begin{matrix}Y_{I}k_{i} & = & (1 − μ_{i}^{2})^{− \frac{1}{2}} h_{i} \\ Y_{I}^{♯}h_{i} & = & −(1 − μ_{i}^{2})^{− \frac{1}{2}} k_{i} \end{matrix}  (62)

5.  \{ \begin{matrix}H_{( _{M^{∗} N^{∗}} )}J_{B}g_{i} & = & μ_{i}(1 − μ_{i}^{2})^{− \frac{1}{2}} h_{i} \\ H( \begin{matrix}\overline{M} & ∗ \\ \overline{N} & ∗ \end{matrix} )f_{i} & = & −μ_{i}(1 − μ_{i}^{2})^{− \frac{1}{2}} k_{i} \end{matrix}  (63)

4 State space realizations

We devote this section to the presentation of state space formulas of the various objects under study. We quote the results obtained previously in Fuhrmann and Ober [1993b] and Ober and Fuhrmann [1993] on the state space representations for the J_{B} normalized coprime factor of a bounded real function. We proceed to the formulas for the doubly coprime factors using the BR-controller. This leads to the state space representation for the B-characteristic R_{B}. The constructions of these state space realizations are based on the use of the stabilizing solutions of two Riccati equations. As the B-characteristic R_{B} of the bounded real function G is stable we have two Lyapunov equations associated with the derived state space realization for R_{B}. The solutions of these equations are necessary for reconstructing the bounded real function G from its B-characteristic R_{B}. However these solutions, naturally positivedefinite, are expressible in terms of the stabilizing solutions of the two Riccati equations. With the solutions to the Lyapunov equations at hand we can reconstruct not only G but actually also the normalized coprime factors. This is of importance as these yields state space formulas for the fractional linear transformation that parametrizes the set of all suboptimal Nehari extensions. This theme is picked up in Section 5.

As in previous studies, see Fuhrmann and Ober [1993a] and Fuhrmann [1993], i.e. in the case of stable systems and minimal systems the normalized coprime factorizations, the normalization being with respect to an appropriate metric, play a major role in the construction of the characteristic functions. The same is true here and we proceed next with the presentation of state space formulas for the J_{B}-normalized coprime factors of a bounded real function. For the purpose of this paper, focusing on the Nehari problem, we do not need the general case and may, without loss of generality, restrict ourselves to strictly proper bounded real functions. For this class of functions the state space formulas simplify and are given by the following theorem. This is a specialization of the general result in Fuhrmann and Ober [1993b]. In turn these formulas are extensions of previous similar results obtained by Meyer and Franklin [1987] and Vidyasagar [1988]. Throughout this section the proofs, mostly computational, are omitted.

Theorem 4.1 Let G be a rational, bounded real, strictly proper, transfer function, and let

G = NM^{−1} = \overline{M}^{−1}\overline{N} (64)

be its normalized right and left J_{B}-coprime factorization. Let

G = ( \begin{matrix}A & B \\ C & 0 \end{matrix} )

be a minimal state space realization. Then

1. A state space realization for ( \begin{matrix}M \\ N \end{matrix} ) is given by

( \begin{matrix}M \\ N \end{matrix} ) = ( \begin{matrix}A + BB^{∗}X & B \\ B^{∗}X & I \\ C & 0 \end{matrix} ) (65)

where X is the stabilizing solution of the Bounded Real Algebraic Riccati Equation (BRRE),

A^{∗}X + XA + C^{∗}C + XBB^{∗}X = 0 (66)

2. A state space realization for ( −\overline{N} \overline{M} ) is given by

\begin{matrix}( −\overline{N} \overline{M} ) = \\ ( \begin{matrix}A + ZC^{∗}C & −B & ZC^{∗} \\ C & 0 & I \end{matrix} ) , \end{matrix} (67)

where Z is the stabilizing solution to the Bounded Real Algebraic Riccati Equation (BRRE),

ZA^{∗} + AZ + BB^{∗} + ZC^{∗}CZ = 0 (68)

We point out that the minimal positive definite solutions can be shown to be actually the stabilizing solutions, i.e. the solutions of the Riccati equations for which A + BB^{∗}Y and A + ZC^{∗}C are stable.

Next we state a theorem giving a state space description of the coprime factors of the Bcontroller. These were characterized in Lemma 2.3.

Theorem 4.2 1. Let ( \begin{matrix}U \\ V \end{matrix} ) be any solution of the Bezout equation \overline{M}V − \overline{N}U = I for

which ( \begin{matrix}M & U \\ N & V \end{matrix} ) has McMillan degree n. Then we have the following state space

realization for it

( \begin{matrix}U \\ V \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & BD′ + ZC^{∗} \\ B^{∗}X & D′ \\ C & I \end{matrix} ) (69)

where D′ is arbitrary and X is the stabilizing solution of the Bounded Real Algebraic Riccati Equation (BRRE),

A^{∗}X + XA + C^{∗}C + XBB^{∗}X = 0 (70)

For the bounded real controller’s coprime factors we have

( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) = ( \begin{matrix}A + BB^{∗}X & −ZC^{∗} \\ B^{∗}X & 0 \\ C & I \end{matrix} ) (71)

2. Let ( \overline{V} −\overline{U} ) be any solution of the Bezout equation \overline{V} M − \overline{U}N = I for which

( \begin{matrix}\overline{V} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) has McMillan degree n. Then we have the following state space realiza-

tion for it

( \overline{V} −\overline{U} ) = ( \begin{matrix}A + ZC^{∗}C & −B & ZC^{∗} \\ \overline{D}′C + B^{∗}X & I & \overline{D}′ \end{matrix} ) (72)

where \overline{D}′ is arbitrary and Z is the stabilizing solution of the Bounded Real Algebraic Riccati Equation (BRRE),

ZA^{∗} + AZ + ZC^{∗}CZ + BB^{∗} = 0 (73)

For the bounded real controller’s coprime factors we have

( \overline{V}_{B} −\overline{U}_{B} ) = ( \begin{matrix}A + ZC^{∗}C & −B & ZC^{∗} \\ B^{∗}X & I & 0 \end{matrix} ) (74)

The following theorem is a special case of a result in Ober and Fuhrmann [1993] and it gives a state space representations of the B-characteristic of G.

Theorem 4.3 Let G be a bounded real function, vanishing at ∞, with a minimal state space realization

G = ( \begin{matrix}A & B \\ C & 0 \end{matrix} ) (75)

Let R be its B-characteristic. Then minimal state space realizations of R are given by

R = ( \begin{matrix}A + BB^{∗}X & B \\ −C(I − ZX) & 0 \end{matrix} ) (76)

and

R = ( \begin{matrix}A + ZC^{∗}C & (I − ZX)B \\ −C & 0 \end{matrix} ) (77)

where X and Z are the stabilizing solutions of the Riccati equations

\{ \begin{matrix}A^{∗}X + XA + XBB^{∗}X + C^{∗}C & = & 0 \\ AZ + ZA^{∗} + ZC^{∗}CZ + BB^{∗} & = & 0 \end{matrix} (78)

That the two realizations of R are isomorphic follows from the Bucy relation

(I − ZX)(A + BB^{∗}X) = (A + ZC^{∗}C)(I − ZX). (79)

The next lemma gives a one to one relation between the solutions to the Lyapunov equations associated with the B-characteristic and the stabilizing solutions of the two Riccati equations associated with the bounded real function G. In particular the inverse relations (82) are instrumental in the proof of Theorem 5.3.

Lemma 4.1 Let G be a bounded real function, vanishing at ∞, with a minimal state space realization

G = ( \begin{matrix}A & B \\ C & 0 \end{matrix} ) (80)

and let X, Z be the stabilizing, or alternatively the minimal positive definite solutions of the Riccati equations (66) and (68). Then

1. For the realization

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

of the B-characteristic, the Lyapunov equations

\{ \begin{matrix}AP + P A^{∗} & = & −BB^{∗} \\ A^{∗}Q + QA & = & −C^{∗}C \end{matrix} (81)

have solutions given by

\{ \begin{matrix}P & = & (I − ZX)^{−1}Z = Z(I − XZ)^{−1} \\ Q & = & X(I − ZX) = (I − XZ)X \end{matrix}

where X, Z are the stabilizing solutions to the Riccati equations (66) and (68).

The stabilizing solutions of these Riccati equations are given by

\{ \begin{matrix}X & = & Q(I − P Q)^{−1} = (I − QP)^{1}Q \\ Z & = & (I − P Q)P = P (I − QP) \end{matrix} (82)

2. For the realization

R = ( \begin{matrix}A + ZC^{∗}C & (I − ZX)B \\ −C & 0 \end{matrix} ) = ( \begin{matrix}A & B \\ C & 0 \end{matrix} )

of the B-characteristic, the Lyapunov equations

\{ \begin{matrix}AP + P A^{∗} & = & −BB^{∗} \\ A^{∗}Q + QA & = & −C^{∗}C \end{matrix} (83)

have solutions given by

\{ \begin{matrix}P & = & Z(I − XZ) = (I − ZX)Z \\ Q & = & (I − XZ)^{−1}X = X(I − ZX)^{−1} \end{matrix}

The stabilizing solutions of these Riccati equations are given by

\{ \begin{matrix}X & = & Q(I − P Q) = (I − QP)Q \\ Z & = & (I − P Q)^{−1}P = P (I − QP)^{−1} \end{matrix} (84)

Our next step is the inversion, via state space formulas, of the characteristic map. This is based on Ober and Fuhrmann [1993].

Theorem 4.4 Let G be a bounded real function, vanishing at ∞, Let R be its B-characteristic have a state space realization

R = ( \begin{matrix}A & B \\ C & 0 \end{matrix} ) (85)

Let P and Q be the solutions of the Lyapunov equations

\{ \begin{matrix}AP + P A^{∗} & = & −BB^{∗} \\ A^{∗}Q + QA & = & −C^{∗}C \end{matrix} (86)

Then state space realization of G are given by

G = ( \begin{matrix}A & B \\ C & 0 \end{matrix} ) = ( \begin{matrix}A − BB^{∗}Q(I − P Q)^{−1} & B \\ −C(I − P Q)^{−1} & 0 \end{matrix} ) (87)

and

G = ( \begin{matrix}A & B \\ C & 0 \end{matrix} ) = ( \begin{matrix}A − P (I − QP)^{−1}C^{∗}C & −(I − P Q)^{−1}B \\ C & 0 \end{matrix} ) (88)

We conclude this section by giving state space representations for the J_{B}-normalized coprime factors of G in terms of a minimal realization of the B-characteristic of G.

Theorem 4.5 Let G be a bounded real function, vanishing at ∞, with a minimal state space realization of its B-characteristic given by

R = ( \begin{matrix}A & B \\ C & 0 \end{matrix} )

Let P and Q be the solutions of the Lyapunov equations

\{ \begin{matrix}AP + P A^{∗} & = & −BB^{∗} \\ A^{∗}Q + QA & = & −C^{∗}C \end{matrix} (89)

Let

G = NM^{−1} = \overline{M}^{−1}\overline{N} (90)

be the normalized J_{B}-coprime factorizations of G. Then

1. A state space realization of the right normalized coprime factors is given by

( \begin{matrix}M \\ N \end{matrix} ) = ( \begin{matrix}A & B \\ B^{∗}Q(I − P Q)^{−1} & I \\ −C(I − P Q)^{−1} & 0 \end{matrix} ) (91)

2. A state space realization of the left normalized coprime factors is given by.

( −\overline{N} \overline{M} ) = ( \begin{matrix}A & (I − P Q)^{−1}B & (I − P Q)^{−1}P C^{∗} \\ C & 0 & I \end{matrix} ) (92)

5 Suboptimal Nehari extensions

There is an extensive literature on various aspects of the Nehari extension problem, its applications and its connections to other basic mathematical problems. Nehari’s theorem is very closely associated to the commutant lifting theorem and hence to a whole class of interpolation problems. Another way to look at it is as an optimization problem where, given a function G ∈ L^{∞}(iR), we look for a best approximation for it in H_{+}^{∞} (considered as a subspace of L^{∞}(iR).

There is an immediate lower bound on the best approximation given in terms of the norm of an induced Hankel operator. In fact if the Hankel operator H_{G} : H_{+}^{2} ⟶ H_{−}^{2} is defined by

H_{G}f = P_{−}Gf, f ∈ H_{+}^{2}

then, noting that for every Q ∈ H_{+}^{∞}

∥H_{G}∥ = ∥H_{G−Q}∥ ≤ ∥G − Q∥_{∞}

we have

∥H_{G}∥ ≤ inf_{Q∈H_{+}^{∞}}∥G − Q∥_{∞}.

In fact, the content of Nehari’s theorem is that this lower bound is achievable. In particular, if ∥H_{G}∥ < 1, then there exists a Q ∈ H_{+}^{∞} for which ∥H_{G}∥ = ∥G − Q∥_{∞}. The function G − Q is referred to as an optimal Nehari extension of G. If however, with ∥H_{G}∥ < 1, we look for a Q ∈ H_{+}^{∞} such that ∥G−Q∥_{∞} ≤ 1 then we refer to G−Q as a suboptimal Nehari extension of G. If our starting point is a rational, strictly proper function G ∈ H_{−}^{∞} then such a function Q is called a suboptimal Nehari complement of G.

Our approach to the solution of the suboptimal Nehari extension problem will be based on the observation made after the proof of Lemma 2.5. There we saw that in terms of the NJ_{B}CF of a bounded real function G we immediately can write down a suboptimal Nehari extension of R_{B}^{∗}, the B-characteristic of G. This allows us to reverse the reasoning. If we start with an arbitrary, rational, strictly proper antistable function R^{∗} ∈ H_{−}^{∞}, our first step will be to identify it as the B-characteristic of some unique, strictly proper, bounded real function. This yields at least one solution of the suboptimal Nehari extension problem, given by equation (12). From there it is a short road to the parametrization of all suboptimal solutions.

5.1 Feedback and scattering formalism

Before proceeding with the analysis of the suboptimal Nehari extension problem we find it of use to collect the basic facts concerning the Julia transform which is the tool connecting the feedback and scattering formalisms.

The standard feedback configuration is given by the following diagram.

[figure: G; u; y; v; z]

The corresponding closed loop system, given the dynamic feedback v = Kz is given by

[figure: G; K; u; y; v; z]

This is equivalent to the equations

( \begin{matrix}y \\ z \end{matrix} ) = ( \begin{matrix}G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} ) ( \begin{matrix}u \\ v \end{matrix} ) (93)

and

v = Kz (94) The closed loop transfer function is easily computed to be G_{11} + G_{12}K(I − G_{22}K)^{−1}G_{21} The scattering formalism is obtained by changing our view on the inputs and outputs. This is summed by the following diagram.

[figure: Θ; u; z; y; v]

In turn this is equivalent to

( \begin{matrix}u \\ y \end{matrix} ) = ( \begin{matrix}Θ_{11} & Θ_{12} \\ Θ_{21} & Θ_{22} \end{matrix} ) ( \begin{matrix}z \\ v \end{matrix} ) (95)

Closing the loop by the same feedback as before, namely by v = Kz leads to

[figure: Θ; K; u; z; y; v]

In terms of the scattering function Θ the closed loop transfer function is

(Θ_{21} + Θ_{22}K)(Θ_{11} + Θ_{12}K)^{−1}

. The passage from the feedback to scattering formalism, that is from G to Θ, is given by the Julia transform, namely

Θ = ( \begin{matrix}G_{21}^{−1} & −G_{21}^{−1}G_{22} \\ G_{11}G_{21}^{−1} & G_{12} − G_{11}G_{21}^{−1}G_{22} \end{matrix} ) (96)

The inverse transformation, that is from the scattering to the feedback formalism is given by

G = ( \begin{matrix}Θ_{21}Θ_{11}^{−1} & Θ_{22} − Θ_{21}Θ_{11}^{−1}Θ_{12} \\ Θ_{11}^{−1} & −Θ_{11}^{−1}Θ_{12} \end{matrix} ) (97)

Clearly the Julia transform is dependent on our choice of variables as well as their order. Thus if we would write

( \begin{matrix}z \\ y \end{matrix} ) = ( \begin{matrix}G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} ) ( \begin{matrix}v \\ u \end{matrix} ) (98)

and

( \begin{matrix}u \\ y \end{matrix} ) = ( \begin{matrix}Θ_{11} & Θ_{12} \\ Θ_{21} & Θ_{22} \end{matrix} ) ( \begin{matrix}v \\ z \end{matrix} ) (99)

then the Julia transform would be

Θ = ( \begin{matrix}−G_{12}^{−1}G_{11} & G_{12}^{−1} \\ G_{21} − G_{22}G_{12}^{−1}G_{11} & G_{22}G_{12}^{−1} \end{matrix} ) . (100)

The Julia transform and its inverse preserve McMillan degree. State space formulas for the Julia transform can be found in Gohberg and Rubinstein [1988].

The following is well known, see Potapov [1955].

Theorem 5.1 Assume G = ( \begin{matrix}G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} ) is inner and that G_{21} is invertible. Then, with

J = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) the matrix function Θ = ( \begin{matrix}G_{21}^{−1} & −G_{21}^{−1}G_{22} \\ G_{11}G_{21}^{−1} & G_{12} − G_{11}G_{21}^{−1}G_{22} \end{matrix} ) is J-inner.

5.2 Parametrization

We begin this section with the study of a special all-pass function, defined in the following theorem.

Theorem 5.2 Let G be a bounded real function of McMillan degree n and let

G = NM^{−1} = \overline{M}^{−1}\overline{N} be NJ_{B}CF. Then

1. ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) is an all-pass function of McMillan degree 2n.

2. The Julia transform of ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) is ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ).

3. We have the following additive decomposition

( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) = ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −M^{−1}U_{B} \end{matrix} ) + ( \begin{matrix}0 & 0 \\ 0 & R_{B}^{∗} \end{matrix} )

4. The McMillan degrees of ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −M^{−1}U_{B} \end{matrix} ) and ( \begin{matrix}0 & 0 \\ 0 & R_{B}^{∗} \end{matrix} ) are both equal to n.

5. The singular values of the involuted Hankel operator Ĥ_{( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −M^{−1}U_{B} \end{matrix} )} are equal

to the singular values of H_{R_{B}^{∗}}.

6. The Julia transform of ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −M^{−1}U_{B} \end{matrix} ) is ( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ).

Proof:

1. We compute, using the equalities \overline{M}^{∗}\overline{M} − NN^{∗} = I and M^{∗}M − N^{∗}N = I,

\begin{matrix}( \begin{matrix}M^{−∗}N^{∗} & M^{−∗} \\ \overline{M}^{−∗} & \overline{M}^{−∗}N \end{matrix} ) & ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) \\ = ( \begin{matrix}M^{−∗}(N^{∗}N + I)M^{−1} & 0 \\ 0 & \overline{M}^{−∗}(NN^{∗} + I)\overline{M}^{−1} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) \end{matrix}

2. By direct computation, using equation (96).

3. Follows from equation (12).

4. We know that the McMillan degrees of both ( \begin{matrix}M \\ N \end{matrix} ) and ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) are equal to n,

hence the McMillan degree of ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) is equal to 2n. Now the Julia transform

preserves McMillan degree so the McMillan degree of ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) is equal

to 2n. Since the B-characteristic map preserves McMillan degree, we must have that the McMillan degree of R_{B}^{∗} is equal to n, and the result follows.

5. Follows from Theorem 1.6 in Appendix 5 of Nikolskii [1985].

6. By direct computation, using the Julia transform equation (96) and the Bezout equation \overline{M}V − \overline{N}U = I.

We pass on now to the parametrization of all suboptimal Nehari extensions of a rational, strictly proper, antistable function R^{∗} ∈ H_{−}^{∞}.

Theorem 5.3 Let R^{∗} ∈ H_{−}^{∞} be a rational, strictly proper, antistable function with ∥H_{R^{∗}}∥ < 1. Let G be the bounded real function obtained as the inverse B-characteristic of R^{∗} and let

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

be NJ_{B}CF of G. Then the set

\{−(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}|Q ∈ BH_{+}^{∞}\} (101)

is a parametrization of the set of all suboptimal Nehari extensions of R^{∗}.

Proof: Since all the Hankel singular values of R^{∗} are less than 1 we can consider it to be the B-characteristic function of some, uniquely determined, strictly proper H_{+}^{∞} contraction G. We consider now the NJ_{B}CF of G, i.e.

G = NM^{−1} = \overline{M}^{−1}\overline{N} (102)

and go back to the all-pass function

( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) (103)

We consider it as the transfer function of the two port

[figure: G; Q; u; y; v; z]

and write this time

( \begin{matrix}z \\ y \end{matrix} ) = ( \begin{matrix}G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} ) ( \begin{matrix}v \\ u \end{matrix} ) (104)

which we terminate by

v = −Qz (105)

The closed loop transfer function is easily computed to be G_{22} − G_{21}Q(I + G_{11}Q)^{−1}G_{12}.

For the particular choice Q = 0, we get −N^{∗}\overline{M}^{−1}. As ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) is an all-pass

function, i.e. it is pointwise unitary on the imaginary axis, then for every Q of norm ≤ 1 the transfer function G_{22} − G_{21}Q(I + G_{11}Q)^{−1}G_{12} is contractive. This we compute to be

\begin{matrix}−N^{∗}\overline{M}^{−1} − M^{−1}Q(I + NM^{−1}Q)^{−1}\overline{M}^{−1} \\ = −M^{−1}\overline{N}^{∗} − M^{−1}Q(I + \overline{M}^{−1}\overline{N}Q)^{−1}\overline{M}^{−1} \\ = −M^{−1}\overline{N}^{∗} − M^{−1}Q(\overline{M} + \overline{N}Q)^{−1} \\ = −M^{−1}(\overline{N}^{∗}(\overline{M} + \overline{N}Q) + Q)(\overline{M} + \overline{N}Q)^{−1} \\ = −M^{−1}(\overline{N}^{∗}\overline{M} + \overline{N}^{∗}\overline{N}Q + Q)(\overline{M} + \overline{N}Q)^{−1} \\ = −M^{−1}(\overline{N}^{∗}\overline{M} + MM^{∗}Q)(\overline{M} + \overline{N}Q)^{−1} \\ = −M^{−1}(MN^{∗} + MM^{∗}Q)(\overline{M} + \overline{N}Q)^{−1} \\ = −(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1} \end{matrix} (106)

or

N^{∗}\overline{M}^{−1} + M^{−1}Q(I + NM^{−1}Q)^{−1}\overline{M}^{−1} = (N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}. (107)

Therefore

N^{∗}\overline{M}^{−1} − (N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1} = −M^{−1}Q(I + NM^{−1}Q)^{−1}\overline{M}^{−1} (108)

which is clearly in H_{+}^{∞} whenever Q ∈ H_{+}^{∞} is contractive. Thus for each contraction Q ∈ H_{+}^{∞} the function (N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1} is a Nehari extension of R^{∗}.

Note that the previous computation is basically the computation of the Julia transform, i.e. we write

( \begin{matrix}u \\ y \end{matrix} ) = ( \begin{matrix}Θ_{11} & Θ_{12} \\ Θ_{21} & Θ_{22} \end{matrix} ) ( \begin{matrix}v \\ z \end{matrix} ) (109)

terminated by v = −Qz. Again a standard computation leads to

Θ = ( \begin{matrix}−G_{12}^{−1}G_{11} & G_{12}^{−1} \\ G_{21} − G_{22}G_{12}^{−1}G_{11} & G_{22}G_{12}^{−1} \end{matrix} ) = ( \begin{matrix}−\overline{N} & \overline{M} \\ M^{∗} & −N^{∗} \end{matrix} ) (110)

Here we used the fact that M^{−1} + N^{∗}\overline{M}^{−1}\overline{M}NM^{−1} = (I + N^{∗}N)M^{−1} = M^{∗}. So with the termination z = −Qv, we have

y = −(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}u

and of course with Q = 0 we get the transfer function −N^{∗}\overline{M}^{−1}. This particular Nehari extension will be referred to as the central Nehari extension.

Next we show that

\{−(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}|Q ∈ BH_{+}^{∞}\} (111)

exhausts the set of all Nehari extensions. To see this let K ∈ L^{∞} be any suboptimal Nehari extension of R^{∗}. We set

Q = −(M^{∗} + K\overline{N})^{−1}(N^{∗} + K\overline{M}) = −(\overline{N}^{∗} + MK)(\overline{M} + NK)^{−1}

Since ∥K∥_{∞} ≤ 1 it follows, by the fact that ( \begin{matrix}N^{∗} & \overline{M} \\ M^{∗} & \overline{N} \end{matrix} ) is J-unitary, that Q is also in the

unit ball of L^{∞}. As the previous fractional linear transformation can be inverted, we can write

K = −(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1} = −(M + QN)^{−1}(\overline{N}^{∗} + Q\overline{M}^{∗}).

So it remains to show that Q ∈ H_{+}^{∞}. As both −N^{∗}\overline{M}^{−1} = −M^{−1}\overline{N}^{∗} and K are Nehari extensions of R^{∗}, we have

−N^{∗}\overline{M}^{−1} − K = −N^{∗}\overline{M}^{−1} + (N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}
= −M^{−1}\overline{N}^{∗} + (N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}
= −M^{−1} [\overline{N}^{∗}(\overline{M} + \overline{N}Q) − M(N^{∗} + M^{∗}Q)] (\overline{M} + \overline{N}Q)^{−1}
= −M^{−1} [(\overline{N}^{∗}\overline{M} − MN^{∗}) + (\overline{N}^{∗}\overline{N} − MM^{∗})Q] (\overline{M} + \overline{N}Q)^{−1}
= M^{−1}Q(\overline{M} + \overline{N}Q)^{−1}
= M^{−1}Q(I + \overline{M}^{−1}\overline{N}Q)^{−1}\overline{M}^{−1}
= M^{−1}Q(I + GQ)^{−1}\overline{M}^{−1} ∈ H_{+}^{∞}.

Since both M and \overline{M} are invertible in H_{+}^{∞} it follows that Q(I + GQ)^{−1} ∈ H_{+}^{∞}. The fact that G ∈ H_{+}^{∞} implies GQ(I + GQ)^{−1} ∈ H_{+}^{∞}, (I + GQ)^{−1} = I − GQ(I + GQ)^{−1} ∈ H_{+}^{∞} and (I − GQ)(I + GQ)^{−1} ∈ H_{+}^{∞}. Now, as GQ is contractive (I − GQ)(I + GQ)^{−1} is positive real. So I + (I − GQ)(I + GQ)^{−1} = 2(I + GQ)^{−1} is strictly positive real and hence invertible in H_{+}^{∞}. Thus we conclude that Q = Q(I + GQ)^{−1}(I + GQ) ∈ H_{+}^{∞}. This completes the proof.

Thus the solvability of the problem of parametrizing all suboptimal Nehari extensions is seen

to be through a special type of Nehari extension, namely the extension of ( \begin{matrix}0 & 0 \\ 0 & R_{B}^{∗} \end{matrix} ) to an

all-pass function. In a sense this seems to be the dual to the standard four block problem in H^{∞}-control theory. The further clarification of this connection is worthy of further study.

Theorem 5.3 is of interest only if, starting with a rational, strictly proper, antistable function R^{∗} ∈ H_{−}^{∞} with ∥H_{R^{∗}}∥ < 1, we can find the J_{B}-normalized coprime factors of a bounded real function G obtained as the inverse B-characteristic of R^{∗}. This problem we solved in Theorem 4.4, inverting the B-characteristic function using state space methods. In fact we can do better than that, for we have derived, in Theorem 4.5, state space formulas for the normalized coprime factors of G directly in terms of a minimal realization of R. Thus we can state.

Corollary 5.1 Let R^{∗} ∈ H_{−}^{∞} be a rational, strictly proper, antistable function with ∥H_{R^{∗}}∥ < 1. Let

R = ( \begin{matrix}A & B \\ C & 0 \end{matrix} ) (112)

be a minimal realization. Let P and Q be the solutions of the Lyapunov equations

\{ \begin{matrix}AP + P A^{∗} & = & −BB^{∗} \\ A^{∗}Q + QA & = & −C^{∗}C \end{matrix} (113)

Define

( \begin{matrix}M \\ N \end{matrix} ) = ( \begin{matrix}A & B \\ B^{∗}Q(I − P Q)^{−1} & I \\ −C(I − P Q)^{−1} & 0 \end{matrix} ) (114)

and

( −\overline{N} \overline{M} ) = ( \begin{matrix}A & (I − P Q)^{−1}B & (I − P Q)^{−1}P C^{∗} \\ C & 0 & I \end{matrix} ) (115)

Then the set

\{−(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}|Q ∈ BH_{+}^{∞}\} (116)

is a parametrization of the set of all suboptimal Nehari extensions of R^{∗}.

These formulas can be found in Francis [1987]. That exposition is based in turn on Ball and Helton [1983] and Ball and Ran [1986].

In the next two sections we present two more methods for this inversion, one based on J_{B}- spectral factorizations, the other a method of solving a Hankel operator based system of equations.

5.3 Connection to J spectral factorizations

We explore now the connections to the J-spectral factorization approach to the problem of parametrizing all suboptimal Nehari extensions.

We observed already the importance of the additive decomposition

( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) = ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −\overline{U}_{B}\overline{M}^{−1} \end{matrix} ) + ( \begin{matrix}0 & 0 \\ 0 & R_{B}^{∗} \end{matrix} ) . (117)

The Julia transform of ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −N^{∗}\overline{M}^{−1} \end{matrix} ) was computed to be ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) whereas

that of ( \begin{matrix}NM^{−1} & \overline{M}^{−1} \\ M^{−1} & −\overline{U}_{B}\overline{M}^{−1} \end{matrix} ) is ( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ). Here U, V correspond to the BR controller.

It is of interest to compute the difference of the two Julia transforms. Recall that we have the relation

( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) = ( \begin{matrix}U_{B} \\ V_{B} \end{matrix} ) − ( \begin{matrix}M \\ N \end{matrix} ) R_{B}^{∗} (118)

This leads to

( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) − ( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ) = − ( \begin{matrix}0 & MR^{∗} \\ 0 & NR^{∗} \end{matrix} ) (119)

or

( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) = ( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ) ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) . (120)

Equivalently,

( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ) = ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & R^{∗} \\ 0 & I \end{matrix} ) . (121)

Note that ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) is J-unitary whereas ( \begin{matrix}I & R^{∗} \\ 0 & I \end{matrix} ) . is a unit in H_{−}^{∞}, therefore this

provides an analogue of inner-outer factorizations.

Equation (121) can be rewritten, using the equality

( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} )

or an equivalent one derived from it, namely

( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} )

as

( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) = ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) . (122)

This leads to

\begin{matrix}( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) & ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) & ( \begin{matrix}I & 0 \\ −R & I \end{matrix} ) \\ = ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}\overline{V} _{B}^{∗} & −\overline{N}^{∗} \\ −\overline{U}_{B}^{∗} & \overline{M}^{∗} \end{matrix} ) \\ = ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}\overline{V} _{B}^{∗} & −\overline{N}^{∗} \\ −\overline{U}_{B}^{∗} & \overline{M}^{∗} \end{matrix} ) \end{matrix} (123)

So ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ∈ H_{+}^{∞} can be obtained from ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) by a process of spectral factorization. Thus we can state.

Theorem 5.4 Let R^{∗} ∈ H_{−}^{∞} be a rational, strictly proper, antistable function with ∥H_{R^{∗}}∥ < 1. Let Γ = ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) ∈ H_{−}^{∞} and let

Γ_{+} = ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ∈ H_{+}^{∞} (124)

be the solution to the J_{B}-spectral factorization problem

\{ \begin{matrix}Γ_{+}J_{B}Γ_{+}^{∗} & = & ΓJ_{B}Γ^{∗} \\ Γ_{+}(∞) & = & ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) . \end{matrix} (125)

Defining the H_{+}^{∞} functions M, N, U, V by

Γ_{+}^{−1} = ( \begin{matrix}M & U \\ N & V \end{matrix} ) (126)

then the set

\{−(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}|Q ∈ BH_{+}^{∞}\} (127) is a parametrization of the set of all suboptimal Nehari extensions of R^{∗}.

Proof: We have to show that if Γ_{+} = ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) is the solution to the J_{B}-spectral fac-

torization problem there exists abounded real function G whose normalized coprime factors are \overline{M}, \overline{N}, whose B-controller coprime factors are \overline{U}, \overline{V} and whose B-characteristic is R^{∗}.

So assume now that Γ_{+} = ( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) is the unique solution to the J_{B} spectral factor-

ization problem (125). Obviously, in view of (126), we have the doubly coprime factorization

( \begin{matrix}\overline{V}_{B} & −\overline{U}_{B} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & U \\ N & V \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

Now, with L = Γ_{+}^{−1}Γ, we have

\begin{matrix}( \begin{matrix}L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix} ) & = & ( \begin{matrix}M & U \\ N & V \end{matrix} ) ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) \\ & = & ( \begin{matrix}M & U − MR^{∗} \\ N & V − NR^{∗} \end{matrix} ) . \end{matrix} (128)

This implies L_{11} = M and L_{21} = N. Similarly, using the equalities L = Γ_{+}^{−1}Γ and L^{∗}J_{B}L = LJ_{B}L^{∗} = J_{B}, we have L^{∗} = Γ^{∗}Γ_{+}^{−∗} so

J_{B} = LJ_{B}L^{∗} = LJ_{B}Γ^{∗}Γ_{+}^{−∗}

This implies

L = J_{B}Γ_{+}^{∗}Γ^{−∗}J_{B}

= ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}\overline{V}^{∗} & −\overline{N}^{∗} \\ −\overline{U}^{∗} & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ R & I \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} )

= ( \begin{matrix}\overline{V}^{∗} & −\overline{N}^{∗} \\ \overline{U}^{∗} & −\overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ R & −I \end{matrix} )

which implies L_{12} = \overline{N}^{∗} and L_{22} = \overline{M}^{∗}. Thus J_{B} = LJ_{B}L^{∗} = L^{∗}J_{B}L implies the identities

( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) (129)

and

( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) (130)

We define now G = NM^{−1} = \overline{M}^{−1}\overline{N}. Obviously G is bounded real and UV^{−1} = \overline{V}^{−1}\overline{U} is a stabilizing controller. Since from (128) we have

( \begin{matrix}U \\ V \end{matrix} ) = ( \begin{matrix}M \\ N \end{matrix} ) \overline{R}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )

we get

M^{∗}U − N^{∗}V = (M^{∗}M − N^{∗}N)R^{∗} + (M^{∗}\overline{N}^{∗} − N^{∗}\overline{M}^{∗}) = R^{∗} ∈ H_{−}^{∞},

and R^{∗} strictly proper. So U, V and \overline{U}, \overline{V} are the right and left coprime factors respectively of the BR controller of G. In particular R^{∗} is the B-characteristic of G. We can apply now Theorem 5.3 to complete the proof.

5.4 Connection to Krein and Melik-Adamyan Method

We outline now the Krein and Melik-Adamyan [1984a,b] approach, another close exposition is Dym [1989], to the problem of parametrization of all suboptimal Nehari complements. We will do so by showing that soving the previous J_{B}-spectral factorization problem is equivalent to solving a special, Hankel based, system of equations.

Theorem 5.5 Let R^{∗} ∈ H_{−}^{∞} be rational and strictly proper and satisfy ∥H_{R^{∗}}∥ < 1. Let

Γ = ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) . (131)

Then Γ_{+} ∈ H_{+}^{∞} is a solution to the J_{B}-spectral factorization problem

\{ \begin{matrix}ΓJ_{B}Γ^{∗} & = & Γ_{+}J_{B}Γ_{+}^{∗} \\ Γ_{+}(∞) & = & ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) \end{matrix} (132)

if and only if Γ = Γ_{+}L with

L = ( \begin{matrix}L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix} ) = ( \begin{matrix}I + Φ_{+} & Φ_{−} \\ \overline{Φ}_{+} & I + \overline{Φ}_{−} \end{matrix} ) = ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) (133)

and Φ_{+}, Φ_{−} and \overline{Φ}_{+}, \overline{Φ}_{−} are the, unique, solutions to the systems of equations

\{ \begin{matrix}Φ_{−}^{∗} + H_{R^{∗}}^{∗}Φ_{+}^{∗} & = & −R \\ Φ_{+}^{∗} + H_{R^{∗}}Φ_{−}^{∗} & = & 0 \end{matrix} (134)

and

\{ \begin{matrix}\overline{Φ}_{−}^{∗} + H_{R^{∗}}^{∗}\overline{Φ}_{+}^{∗} & = & 0 \\ \overline{Φ}_{+}^{∗} + H_{R^{∗}} \overline{Φ}_{−}^{∗} & = & −R^{∗} \end{matrix} (135)

Proof: Assume first R^{∗} ∈ H_{−}^{∞} and ∥H_{R^{∗}}∥ < 1, and Γ is defined by (131). We proceed with solving the J_{B}-spectral factorization problem (132). Since

(Γ_{+}^{−1}Γ)J_{B}(Γ^{∗}Γ_{+}^{−∗}) = J_{B} (136)

it follows that, with L defined by

L = Γ_{+}^{−1}Γ, (137)

that

LJ_{B}L^{∗} = J_{B}. (138)

By our normalization of Γ_{+} we have L(∞) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ).

We set now

Γ_{+} = ( \begin{matrix}K_{11} & K_{12} \\ K_{21} & K_{22} \end{matrix} ) (139)

with K_{ij} ∈ H_{+}^{∞}. We also set

Γ_{+}^{−1} = ( \begin{matrix}H_{11} & H_{12} \\ H_{21} & H_{22} \end{matrix} ) (140)

and again H_{ij} ∈ H_{+}^{∞}.

Since L = Γ_{+}^{−1}Γ we have

( \begin{matrix}H_{11} & H_{12} \\ H_{21} & H_{22} \end{matrix} ) ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) = ( \begin{matrix}H_{11} & H_{12} − H_{11}R_{B}^{∗} \\ H_{21} & H_{22} − H_{21}R_{B}^{∗} \end{matrix} ) = ( \begin{matrix}L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix} )

This implies the inclusions

L_{11}, L_{21} ∈ H_{+}^{∞}. (141)

Next note that as, LJ_{B}L^{∗} = J_{B}, it follows that

L^{−∗} = J_{B}LJ_{B}. (142)

Also Γ = Γ_{+}L implies Γ^{∗} = L^{∗}Γ_{+}^{∗} and hence Γ_{+}^{∗} = L^{−∗}Γ^{∗} = (J_{B}LJ_{B})Γ^{∗} and hence

\begin{aligned}( \begin{matrix}K_{11}^{∗} & K_{21}^{∗} \\ K_{12}^{∗} & K_{22}^{∗} \end{matrix} ) & = & ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}I & 0 \\ −R & I \end{matrix} ) \\ & = & ( \begin{matrix}L_{11} & −L_{12} \\ −L_{21} & L_{22} \end{matrix} ) ( \begin{matrix}I & 0 \\ −R & I \end{matrix} ) \end{aligned}

Thus we get

L_{12}, L_{22} ∈ H_{−}^{∞} (143)

Similarly

( \begin{matrix}H_{11} & H_{12} \\ H_{21} & H_{22} \end{matrix} ) ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) = ( \begin{matrix}L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix} ) (144)

we have

( \begin{matrix}I & 0 \\ −R & I \end{matrix} ) ( \begin{matrix}H_{11}^{∗} & H_{21}^{∗} \\ H_{12}^{∗} & H_{22}^{∗} \end{matrix} ) = ( \begin{matrix}L_{11}^{∗} & L_{21}^{∗} \\ L_{12}^{∗} & L_{22}^{∗} \end{matrix} ) (145)

or

( \begin{matrix}H_{11}^{∗} & H_{21}^{∗} \\ H_{12}^{∗} & H_{22}^{∗} \end{matrix} ) = ( \begin{matrix}I & 0 \\ R & I \end{matrix} ) ( \begin{matrix}L_{11}^{∗} & L_{21}^{∗} \\ L_{12}^{∗} & L_{22}^{∗} \end{matrix} ) (146)

In particular we have the following equalities

\{ \begin{matrix}H_{12}^{∗} & = & RL_{11}^{∗} + L_{12}^{∗} \\ H_{22}^{∗} & = & RL_{21}^{∗} + L_{22}^{∗} \end{matrix} (147)

We apply the orthogonal projection P_{+} to get

\begin{matrix}P_{+}H_{12}^{∗} = P_{+}R(L_{11}^{∗} − L_{11}(∞)^{∗}) + L_{12}^{∗} + P_{+}RL_{11}(∞)^{∗} = 0 \\ P_{+}(H_{22}^{∗} − H_{22}(∞)^{∗}) = P_{+}RL_{21}^{∗} + P_{+}(L_{22}^{∗} − L_{22}(∞)^{∗}) \end{matrix} (148)

On the other hand

( \begin{matrix}K_{11} & K_{12} \\ K_{21} & K_{22} \end{matrix} ) = ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) ( \begin{matrix}L_{11}^{∗} & −L_{21}^{∗} \\ −L_{12}^{∗} & L_{22}^{∗} \end{matrix} ) (149)

we obtain

\{ \begin{matrix}K_{11} & = & L_{11}^{∗} + R^{∗}L_{12}^{∗} \\ K_{12} & = & −L_{21}^{∗} + R^{∗}L_{22}^{∗} \end{matrix} (150)

and applying the orthogonal projection P_{−} we get

\{ \begin{matrix}P_{−}(L_{11}^{∗} − I) + H_{R^{∗}}L_{12}^{∗} & = & 0 \\ −L_{21}^{∗} − P_{−}R^{∗}(L_{22}^{∗} − I) − R^{∗} & = & 0 \end{matrix} (151)

We set now

\begin{matrix}Φ_{+} & = & L_{11} − I, & Φ_{−} & = & L_{12} \\ \overline{Φ}_{+} & = & L_{21}, & \overline{Φ}_{−} & = & L_{22} − I \end{matrix} (152)

Obviously, Φ_{+}, \overline{Φ}_{+} ∈ H_{+}^{∞} ∩ H_{+}^{2} and Φ_{−}, \overline{Φ}_{−} ∈ H_{−}^{∞} ∩ H_{−}^{2}.

In this notation the equations (148) and (151) can be rewritten together as

\{ \begin{matrix}Φ_{−}^{∗} + H_{R^{∗}}^{∗}Φ_{+}^{∗} & = & −R \\ \overline{Φ}_{−}^{∗} + H_{R^{∗}}^{∗}\overline{Φ}_{+}^{∗} & = & 0 \\ \overline{Φ}_{+}^{∗} + H_{R^{∗}} \overline{Φ}_{−}^{∗} & = & −R^{∗} \\ Φ_{+}^{∗} + H_{R^{∗}}Φ_{−}^{∗} & = & 0 \end{matrix} (153)

We split equations (153) into the two systems of equations (134) and (135). Obviously, by our assumption that ∥H_{R^{∗}}∥ < 1, these equations have a unique solution. We define now

\{ \begin{matrix}M & = & I + Φ_{+}, & N & = & \overline{Φ}_{+} \\ \overline{M} & = & I + \overline{Φ}_{−}^{∗}, & \overline{N} & = & Φ_{−}^{∗} \end{matrix} (154)

Therefore

L = ( \begin{matrix}L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix} ) = ( \begin{matrix}I + Φ_{+} & Φ_{−} \\ \overline{Φ}_{+} & I + \overline{Φ}_{−} \end{matrix} ) = ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) (155)

Since LJ_{B}L^{∗} = J_{B}, we have

( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) (156)

and hence also

( \begin{matrix}M^{∗} & N^{∗} \\ \overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) (157)

This implies the identities

\{ \begin{matrix}M^{∗}M − N^{∗}N & = & I \\ \overline{M}N − \overline{N}M & = & 0 \\ M^{∗}\overline{N}^{∗} − N^{∗}\overline{M}^{∗} & = & 0 \\ \overline{M}\overline{M}^{∗} − \overline{N}\overline{N}^{∗} & = & I \end{matrix} (158)

and

\{ \begin{matrix}\overline{M}^{∗}\overline{M} − NN^{∗} & = & I \\ \overline{M}^{∗}\overline{N} − NM^{∗} & = & 0 \\ \overline{N}^{∗}\overline{M} − MN^{∗} & = & 0 \\ MM^{∗} − \overline{N}^{∗}\overline{N} & = & I \end{matrix} (159)

From this we deduce that for G defined by

G = NM^{−1} = \overline{M}^{−1}\overline{N} (160)

is a bounded real function and these are J_{B} normalized coprime factorization of it.

T = N^{∗}\overline{M}^{−1} = M^{−1}\overline{N}^{∗} (161)

is an L^{∞} contraction.

We will show now that χ_{B}(G) = R^{∗}. To this end we define matrix functions U, V and \overline{U}, \overline{V} by

( \begin{matrix}U \\ V \end{matrix} ) = ( \begin{matrix}M \\ N \end{matrix} ) \overline{R}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) (162)

and

( \overline{V} \overline{U} ) = R^{∗} ( \overline{N} \overline{M} ) + ( M^{∗} N^{∗} ) (163)

respectively. Equations (134) and (135) imply that all these functions are in H_{+}^{∞}. Next we compute

( \overline{V} \overline{U} ) ( \begin{matrix}M \\ −N \end{matrix} ) = [R^{∗} ( \overline{N} \overline{M} ) + ( M^{∗} N^{∗} )] ( \begin{matrix}M \\ −N \end{matrix} ) = I,

and

( −\overline{N} \overline{M} ) ( \begin{matrix}U \\ V \end{matrix} ) = ( −\overline{N} \overline{M} ) [( \begin{matrix}M \\ N \end{matrix} ) \overline{R}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )] = I.

Moreover

( \overline{V} −\overline{U} ) ( \begin{matrix}U \\ V \end{matrix} ) = [R^{∗} ( \overline{N} −\overline{M} ) + ( M^{∗} −N^{∗} )] [( \begin{matrix}M \\ N \end{matrix} ) \overline{R}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )] = 0.

Since

M^{∗}U − N^{∗}V = ( M^{∗} −N^{∗} ) [( \begin{matrix}M \\ N \end{matrix} ) \overline{R}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} )] = R^{∗} ∈ H_{−}^{∞}

and

\overline{U}\overline{M}^{∗} − \overline{V} \overline{N}^{∗} = [R^{∗} ( \overline{N} \overline{M} ) + ( M^{∗} N^{∗} )] ( \begin{matrix}−\overline{N}^{∗} \\ \overline{M} & ∗ \end{matrix} ) = R^{∗},

it follows that R^{∗} is the B-charcteristic of G.

Conversely, assume that Φ_{+}, Φ_{−}, \overline{Φ}_{+}, \overline{Φ}_{−} are the solutions to the system of equations (134) and (134) . We define the H_{+}^{∞} functions N, M, \overline{N}, \overline{M} via (154). Obviously N, \overline{N} are strictly proper, whereas M, \overline{M} are proper and satisfy M(∞) = I and \overline{M}(∞) = I. We show now that

L = ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} )

satisfies L^{∗}J_{B}L = J_{B}.

To this end we rewrite the equation Φ_{−}^{∗} + H_{R^{∗}}^{∗}Φ_{+}^{∗} = −R as \overline{N} + P_{+}R(M^{∗} − I) + R = P_{+}(\overline{N} + RM^{∗}) = 0, i.e. \overline{N} + RM^{∗} ∈ H_{−}^{∞}. Hence, going through similar computations, we get that the functions defined by

\{ \begin{matrix}U & = & MR^{∗} + \overline{N}^{∗} \\ V & = & NR^{∗} + \overline{M}^{∗} \\ \overline{U} & = & R^{∗}\overline{M} + N^{∗} \\ \overline{V} & = & R^{∗}\overline{N} + M^{∗} \end{matrix} (164)

are all in H_{+}^{∞}.

Note that these equations can be written also as

( \begin{matrix}U \\ V \end{matrix} ) = ( \begin{matrix}M \\ N \end{matrix} ) \overline{R}^{∗} + ( \begin{matrix}\overline{N} & ∗ \\ \overline{M} & ∗ \end{matrix} ) (165)

and

( \overline{V} \overline{U} ) = R^{∗} ( \overline{N} \overline{M} ) + ( M^{∗} N^{∗} ) (166)

These equations imply also the following

( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ) = ( \begin{matrix}M & \overline{N}^{∗} \\ N & \overline{M}^{∗} \end{matrix} ) ( \begin{matrix}I & R^{∗} \\ 0 & I \end{matrix} ) . (167)

and

( \begin{matrix}\overline{V}_{B} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) = ( \begin{matrix}I & −R^{∗} \\ 0 & I \end{matrix} ) ( \begin{matrix}M^{∗} & −N^{∗} \\ −\overline{N} & \overline{M} \end{matrix} ) . (168)

Now we can write

MM^{∗} − \overline{N}^{∗}\overline{N} = (M + \overline{N}^{∗}R)M^{∗} − \overline{N}^{∗}(\overline{N} + RM^{∗}) ∈ H_{−}^{∞}

as all factors are in H_{−}^{∞}. Since MM^{∗} − \overline{N}^{∗}\overline{N} is self adjoint we have also MM^{∗} − \overline{N}^{∗}\overline{N} ∈ H_{+}^{∞}, and hence it must be a constant. Using the strict properness of \overline{N} and the fact that M(∞) = I we get

MM^{∗} − \overline{N}^{∗}\overline{N} = I.

Similarly

\overline{M}^{∗}\overline{M} − NN^{∗} = \overline{M}^{∗}(\overline{M} + RN^{∗}) − (N + \overline{M}^{∗}R)N^{∗} ∈ H_{−}^{∞}.

Again this implies

\overline{M}^{∗}\overline{M} − NN^{∗} = I.

Next we write

\overline{M}^{∗}\overline{N} − NM^{∗} = (V − NR^{∗})\overline{N} − N(\overline{V} − R^{∗}\overline{N}) = V \overline{N} − N\overline{V} ∈ H_{+}^{∞}.

On the other hand

\overline{M}^{∗}\overline{N} − NM^{∗} = \overline{M}^{∗}(\overline{N} + RM^{∗}) − (N + \overline{M}^{∗}R)M^{∗} = \overline{M}^{∗}U^{∗}) − \overline{U}^{∗}M^{∗} ∈ H_{−}^{∞}.

These two inclusions imply that \overline{M}^{∗}\overline{N} − NM^{∗} is a constant, and evaluating at ∞ we get

\overline{M}^{∗}\overline{N} − NM^{∗} = 0,

and also

\overline{N}^{∗}\overline{M} − MN^{∗} = 0.

Thus we have derived the identities (159). This is equivalent to L^{∗}J_{B}L = J_{B} and therefore also to LJ_{B}L^{∗} = J_{B}.

We set now

Γ_{+} = ΓL^{−1}.

Clearly we have Γ_{+} is proper and Γ_{+}(∞) = I. Moreover we compute

\begin{aligned}ΓJ_{B}Γ^{∗} & = & Γ_{+}LJ_{B}L^{∗}Γ_{+}^{∗} \\ & = & Γ_{+}J_{B}Γ_{+}^{∗} \end{aligned}

So Γ_{+} as defined is the solution to the J_{B} spectral factorization problem

\begin{matrix}Γ_{+}J_{B}Γ_{+}^{∗} & = & ΓJ_{B}Γ^{∗} \\ Γ_{+}(∞) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) . \end{matrix} (169)

Note that, using the identities (158) and (159), we have the doubly coprime factorization

( \begin{matrix}\overline{V}_{B} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & U_{B} \\ N & V_{B} \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) .

In view of the previous theorem, we can apply Theorem 5.4 to state.

Theorem 5.6 Let R^{∗} ∈ H_{−}^{∞} be a rational, strictly proper, antistable function with ∥H_{R^{∗}}∥ < 1. Let Φ_{+}, Φ_{−}, \overline{Φ}_{+}, \overline{Φ}_{−} be the solutions of the systems of equations (134) and (135). Let M, N, \overline{M}, \overline{N} be defined by equation (154). Then the set

\{−(N^{∗} + M^{∗}Q)(\overline{M} + \overline{N}Q)^{−1}|Q ∈ BH_{+}^{∞}\} (170)

is a parametrization of the set of all suboptimal Nehari extensions of R^{∗}.

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P. A. Fuhrmann Department of Mathematics Ben-Gurion University of the Negev 84105 Beer Sheva, Israel

MSC: 47A15 47A68 47B35 47B50 47N70 93B36