APPROXIMATION BY ANALYTIC MATRIX FUNCTIONS. THE FOUR BLOCK PROBLEM

VLADIMIR. V. PELLER AND SERGEI R. TREIL

Abstract. We study the problem of finding a superoptimal solution to the four block problem. Given a bounded block matrix function ( \begin{matrix}Φ_{11} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) on the unit

circle the four block problem is to minimize the L^{∞} norm of ( \begin{matrix}Φ_{11} − F & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) over F ∈ H^{∞}. Such a minimizing F (an optimal solution) is almost never unique. We consider the problem to find a superoptimal solution which minimizes not only the supremum of the matrix norms but also the suprema of all further singular values. We give a natural condition under which the superoptimal solution is unique.

1. . Introduction

The problem of approximating a given scalar function φ on the unit circle 𝕋 uniformly by functions analytic in the unit disk 𝔻 has been attracting analysts for a long time (see [Kha], [RSh], [Ne], [AAK1-2], [CJ], [PKh]). It was shown in [Kha] that for a continuous function φ such a best approximation is unique while it is not unique in the general case. Later it turned out that this problem is closely related with Hankel operators. Namely, it was proved by Nehari [Ne] that

dist_{L^{∞}}(φ, H^{∞}) = ∥H_{φ}∥,

where H_{φ} : H^{2} → H_{−}^{2} =^{def} L^{2} ⊖ H^{2} is the Hankel operator with symbol φ defined by

H_{φ}f = ℙ_{−}φf, f ∈ H^{2},

(we denote by ℙ_{+} and ℙ_{−} the orthogonal projections onto H^{2} and H_{−}^{2}). Presently the problem of approximating by analytic functions in L^{∞} is called Nehari’s problem.

We shall also need the notion of a Toeplitz operator. Given φ ∈ L^{∞} the Toeplitz operator T_{φ} : H^{2} → H^{2} is defined by

T_{φ}f = ℙ_{+}φf, f ∈ H^{2}.

1991 Mathematics Subject Classification. 47B35, 93B36.

Both authors are partially supported by NSF grant DMS 9304011. Research at MSRI is supported in part by NSF grant DMS-9022140.

Adamyan, Arov and Krein [AAK1-2] found many interesting connections between Hankel operators and Nehari’s problem. In particular they found a more general condition under which a best approximation is unique: if the essential norm ∥H_{φ}∥_{e} is less than ∥H_{φ}∥, then φ has a unique best approximation. (Recall that for an operator T on Hilbert space

∥T ∥_{e} =^{def} inf{∥T − K∥ : K is compact}).

This sufficient uniqueness condition can easily be reformulated in terms of the function φ itself since

∥H_{φ}∥_{e} = dist_{L^{∞}}(φ, H^{∞} + C)

(see [AAK1-2]). They also found a criterion of uniqueness of a best approximation in terms of the corresponding Hankel operator, and in the case of non-uniqueness parametrized all best approximations (optimal solutions of Nehari’s problem), see [AAK1-2]. However it is not very easy to verify whether a function φ in L^{∞} satisfies the criterion.

Carleson and Jacobs [CJ] studied smoothness properties of the best approximation for smooth functions φ. They proved that if φ belongs to the Hölder–Zygmund class Λ_{α}, α > 0, α ∉ ℤ, then the best approximation also belongs to the same class.

Later in [PKh] more general hereditary properties of the non-linear operator of best approximation were studied. For a large class of function spaces X on 𝕋 it was proved that if φ ∈ X and f is the best approximation by analytic functions, then f ∈ X. Note also that in [PKh] Nehari’s problem was also applied in prediction theory which led to a new approach to the problem of describing stationary processes satisfying various regularity conditions in terms of their spectral densities.

A new wave of interest in Nehari’s problem was caused by the development of H^{∞} control theory where Nehari’s problem plays a central role (see [Fr]). Moreover for the needs of H^{∞} control theory it is important to consider Nehari’s problem for matrix-valued functions: given an n × m matrix function Φ on 𝕋 the problem is to approximate Φ by bounded analytic matrix functions Q in the norm

∥Φ − Q∥_{∞} =^{def} ess sup_{ζ∈𝕋}∥Φ(ζ) − F (ζ)∥,

where ∥ · ∥ on the right-hand side is the norm of the matrix as an operator from ℂ^{m} to ℂ^{n}. However in contrast with the scalar case we have uniqueness of a best approximation only in exceptional cases. Indeed, let

Φ = ( \begin{matrix}z̄ & 0 \\ 0 & \frac{1}{2} z̄ \end{matrix} ) .

Clearly dist_{L^{∞}}(z̄, H^{∞}) = 1 and so ∥Φ − Q∥_{∞} ≥ 1 for any Q ∈ H^{∞}. However it is

easy to see that any function of the form ( \begin{matrix}0 & 0 \\ 0 & q \end{matrix} ) with q ∈ H^{∞}, ∥q∥_{∞} ≤ \frac{1}{2},is a best approximation. Intuitively, however, is clear that the “very best” approximation is the zero matrix function 𝕆.

In [Y] Young suggested imposing the following additional assumptions on approximating functions. Let Ω_{0} be the set of best approximations:

Ω_{0} = {Q ∈ H^{∞} : Q minimizes ess sup_{ζ∈𝕋}∥Φ(ζ) − Q(ζ)∥}.

Define inductively the sets Ω_{j} as follows

Ω_{j} = {Q ∈ Ω_{j−1} : Q minimizes ess sup_{ζ∈𝕋}s_{j}(Φ(ζ) − Q(ζ))}

(for a matrix (or an operator) A the jth singular value s_{j}(A), j ≥ 0, is the distance from A to the set of matrices (operators) of rank at most j, s_{0}(A) =^{def} ∥A∥). Elements of Ω_{min{m,n}−1} are called superoptimal approximations of Φ (or superoptimal solutions of Nehari’s problem). Put

t_{j} =^{def} ess sup_{ζ∈𝕋}s_{j}(Φ(ζ) − Q(ζ)), Q ∈ Ω_{j}.

The numbers t_{j}, 0 ≤ j ≤ min{m, n} − 1, are called the superoptimal singular values. As in the scalar case, t_{0} = ∥H_{Φ}∥ (the Hankel operator H_{Φ} : H^{2}(ℂ^{n}) → H^{2}(ℂ^{m}) is defined in the same way as in the scalar case).

Note that Q is a superoptimal solution of Nehari’s problem if and only if it lexicographically minimizes the sequence {s_{j}^{∞}(Φ − Q)}_{j≥0}, where for a matrix function F on 𝕋

s_{j}^{∞}(F) =^{def} ess sup_{ζ∈𝕋}s_{j}(F (ζ)).

It was proved in [PY1] that for Φ ∈ H^{∞} + C there exists a unique superoptimal approximation Q. The method of the proof in [PY1] is based on certain special factorizations of matrix functions (thematic factorizations) and it is constructive. Later in [T] another method was suggested to establish uniqueness in the H^{∞} + C case which is based on weighted Nehari’s problem.

However in the case when the Hankel operator H_{Φ} is non-compact, there was no analog of the Adamyan–Arov–Krein sufficient condition for uniqueness in the case of matrix functions.

Nehari’s problem is a special case of the so-called four block problem which is one of the most important problems in control theory. Let Φ be a block matrix function of the form

Φ = ( \begin{matrix}Φ_{11} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) .

Here Φ has size m × n, Φ_{11} has size m_{1} × n_{1}, and Φ_{2} has size m_{2} × n_{2}. The four block problem is to minimize

∥( \begin{matrix}Φ_{11} − Q & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} )∥_{∞} ,  (1.1)

over bounded analytic functions Q of size m_{1} × n_{1}. A function Q ∈ H^{∞}(M_{m_{1}n_{1}}) is called an optimal solution of the four block problem if it minimizes the norm (1.1).

The four block problem arises naturally when one considers the following (modelmatching) problem in H^{∞} control. Let F, G_{1} and G_{2} be matrix functions of class H^{∞}. The problem is to minimize

∥F − G_{1}QG_{2}∥_{∞}  (1.2)

over Q ∈ H^{∞} (the sizes of the matrix functions in (1.2) are such that (1.2) is meaningful). Many problems in H^{∞} control reduce to the model-matching problem.

Engineers usually consider the case of continuous (on 𝕋), or even rational functions G_{1} and G_{2} and assume that these functions have constant rank on the boundary. Under this assumption the model-matching problem reduces to the four block problem (1.1), while it reduces to Nehari’s problem only if the matrices G_{2}, G_{1}^{∗} have maximal column rank (the rank equals the number of columns). The assumption on the maximal column rank does not hold for many interesting applied problems, so engineers have to consider the four block problem as well.

In the most general case the model matching problem (1.2) reduces to the four block problem under the assumption that the outer parts of functions G_{1} and G_{2}^{∗} are right invertible in L^{∞} (for continuous functions this is equivalent to fact that they have constant rank on 𝕋). The model mathcing problem reduces to Nehari’s problem if the outer parts of functions G_{1} and G_{2}^{∗} are invertible in L^{∞} (which for continuous functions is equivalent to the above maximal column rank assumption).

By analogy with Hankel operators we define the four block operator Γ_{Φ} : H^{2}(ℂ^{n_{1}})⊕ L^{2}(ℂ^{n_{2}}) → H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}) by

Γ_{Φ} ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) = ℙ^{−}Φ ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) ,

where ℙ^{−} is the orthogonal projection from L^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}) onto H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}). As in the case of Hankel operators the infimum in (1.1) is equal to ∥Γ_{Φ}∥ (see [FT]). The matrix function Φ is called a symbol of the four block operator Γ_{Φ} (a four block operator has many different symbols).

As in the case of Nehari’s problem we can define the sets Ω_{j}:

Ω_{0} = \{Q ∈ H^{∞} : Q minimizes ∥( \begin{matrix}Φ_{11} − F & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} )∥_{∞}\} ,

Ω_{j} = \{Q ∈ Ω_{j−1} : Q minimizes ess sup_{ζ∈𝕋}s_{j} (( \begin{matrix}Φ_{11} − F & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) (ζ))\} .

Q is called a superoptimal solution of the four block problem (1.1) if Q ∈ Ω_{min{m_{1},n_{1}}−1}. We define the superoptimal singular values of the four block problem (1.1) by

t_{j} = ess sup_{ζ∈𝕋}s_{j} (( \begin{matrix}Φ_{11} − Q & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) (ζ)) , F ∈ Ω_{j}.

Clearly, t_{0} = ∥Γ_{Φ}∥.

We say that a Φ is a superoptimal symbol of a four block operator Γ if Γ = Γ_{Φ} and the zero function is a superoptimal solution of the corresponding four block problem.

Using a simple compactness argument one can prove easily that a superoptimal solution always exists. However we cannot expect a sufficient condition for uniqueness of the superoptimal approximation which would be similar to the one found in [PY1] in the case of Nehari’s problem. Indeed it can easily be proved that a four block operator cannot be compact unless Φ_{12}, Φ_{21}, and Φ_{22} are identically equal to zero in which case the four block problem is equivalent to Nehari’s problem.

The main result of the paper (Theorem 2.1) is a sufficient condition for the four block problem to have a unique superoptimal solution. This result can be considered as an analog of the Adamyan–Arov–Krein theorem mentioned above which deals with Nehari’s problem in the scalar case. Note that Theorem 2.1 also gives us a new result for Nehari’s problem in the case when the corresponding Hankel operator is non-compact.

The proof is constructive. We give an algorithm to find the unique superoptimal solution. The algorithm is similar to the one given in [PY1], it reduces the problem to the case of matrix functions of lower size. However the proof is considerably more complicated than in the case of Nehari’s problem with compact Hankel operator.

In Section 3 we describe briefly the method of factorization and diagonalization. We construct important matrix functions V and W which can be considered as analogs of the thematic functions defined in [PY1].

In Section 4 we use the construction of Section 3 to parametrize all optimal solutions of the four block problem. This allows us to reduce the problem of finding superoptimal solutions to the case of matrix functions of lower size.

Section 5 is devoted to the proof of the fact that the matrix functions V_{c} and W_{c} which are submatrices of V and W are left invertible in H^{∞}. This is one of the principal points in the proof of the main result.

In Section 6 we prove another crucial fact for the proof of the main result. Namely, we show that if Φ satisfies the hypotheses of Theorem 2.1, then the lower order four block problem, obtained as a result of parametrization in Section 4, also satisfies the hypotheses of Theorem 2.1. This makes it possible to continue the process and complete the proof of the main result.

In Section 7 we study superoptimal symbols of four block operators. We obtain certain special factorizations of such symbols (thematic factorizations), and define the indices of such factorizations. In the case of Nehari’s problem with compact Hankel operators such factorizations were found in [PY1].

To prove the invariance of indices we introduce in Section 8 the notion of a superoptimal weight for the four block operator. This is an analog of the notion introduced in [T] in the case of compact Hankel operators.

In Section [9] we use superoptimal weights to prove that the sums of the indices in a thematic factorization which correspond to equal superoptimal singular values do not depend on the choice of factorization. In the case of compact Hankel operators this invariance property was proved in [PY2]. Note that as in the case of Nehari’s problem for H^{∞} + C functions if there are equal superoptimal singular values, the indices can depend on the choice of thematic factorization (see [PY2]).

The last section is devoted to inequalities between the superoptimal singular values and the singular values of the four block operator. We obtain an inequality which is new even in the case of Nehari’s problem with compact Hankel operator. It is stronger than the one obtained in [PY2].

Note that in [PY1] hereditary properties of the non-linear operator of superoptimal approximation were studied. It was shown there that for a large class of function spaces X the inclusion Φ ∈ X implies that the superoptimal approximant to Φ also belongs to X. It would be interesting to find analogs of such results in the case of the four block problem. In particular we do not know whether the superoptimal solution of the four block problem (1.1) must belong to a Hölder class Λ_{α} if Φ ∈ Λ_{α}.

Throughout this paper we shall denote by M_{m,n} the space of m × n matrices. We shall use the notation L^{∞}(M_{m,n}) and H^{∞}(M_{m,n}) for the spaces of bounded and bounded analytic functions which take values in M_{m,n}. Sometimes if it does not lead to a confusion, we shall simply write L^{∞} and H^{∞} instead of L^{∞}(M_{m,n}) and H^{∞}(M_{m,n}).

A matrix function Θ ∈ H^{∞} is called inner if Θ(ζ) is isometric for almost all ζ ∈ 𝕋. A matrix function F ∈ H^{∞}(M_{m,n}) is called outer if F H^{2}(ℂ^{n}) is dense in H^{2}(ℂ^{m}). A function F ∈ H^{∞}(M_{m,n}) is called co-outer if the transposed function F^{t} ∈ H^{∞}(M_{n,m}) is outer.

2. . The main result

In this section we state the main result of the paper as well as important corollaries. Let Φ be a matrix function of the form

Φ = ( \begin{matrix}Φ_{11} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) ,  (2.1)

where Φ has size m × n, Φ_{11} has size m_{1} × n_{1}, and Φ_{22} has size m_{2} × n_{2}. Recall that Γ_{Φ} is the four block operator defined in Section 1 and {t_{j}} is the sequence of superoptimal singular values.

The following theorem is the main result of the paper.

Theorem 2.1.. Let Φ be a bounded function of the form (2.1). Suppose that ∥Γ_{Φ}∥_{e} is less than the smallest nonzero superoptimal singular value. Then there exists a unique superoptimal solution Q of the four block problem for Φ. The singular values

s_{j} (( \begin{matrix}Φ_{11} − Q & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) (ζ)) , 1 ≤ j ≤ d − 1,

are constant on 𝕋.

The following partial case of Theorem 2.1 improves the result of [PY1] on uniqueness of superoptimal solutions of Nehari’s problem.

Theorem 2.2.. Let Φ be bounded matrix function on 𝕋. Suppose that ∥H_{Φ}∥_{e} is less that the smallest nonzero superoptimal singular value of Nehari’s problem. Then Φ has a unique superoptimal approximation Q by bounded analytic matrix functions. The singular values s_{j}(Φ(ζ) − Q(ζ)) are constant on 𝕋.

3. . Diagonalization

In this section we start with a maximizing vector of the four block operator and we construct a certain special unitary-valued matrix. This allows us to achieve a diagonalization. Later using this diagonalization we shall reduce the problem to the case of a matrix function of a lower size.

Lemma 3.1.. Let v be an n×1 inner matrix function. Then there exists a co-outer function V_{c} ∈ H^{∞}(M_{n,n−1}) such that the matrix function

V =^{def} ( v \overline{V}_{c} )

is unitary-valued on 𝕋.

In [PY1] a stronger result was obtained. It was shown that all minors of V on the first column are in H^{∞}. This property of analyticity of minors was essential for the proof of the uniqeness of a superoptimal solution of Nehari’s problem which was given in [PY1]. Earlier the existence of a co-outer V_{c} satisfying the requirement of Lemma 3.1 was proved in [Va], however the property of analyticity of minors was not noticed in [Va]. It also can be shown that if we V_{c}^{(1)} and V_{c}^{(2)} are n×(n−1) co-outer functions satisfying the requirements of Lemma 3.1, then there exists a constant unitary matrix U such that V_{c}^{(1)} = V_{c}^{(2)}U (see [Va], [PY1]).

To start the procedure we need a maximizing vector of the four block operator Γ_{Φ}, i.e., a nonzero vector f ∈ H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) such that ∥Γ_{Φ}f∥ = ∥Γ_{Φ}∥ · ∥f∥. If Φ satisfies the hypotheses of Theorem 2.1 and Γ_{Φ} ≠ 𝕆, then ∥Γ_{Φ}∥ = t_{0} > ∥Γ_{Φ}∥_{e} and so a maximizing vector for Γ_{Φ} exists.

The following fact is well-known in the case of Hankel operators (see [AAK3]). The proof of it in the case of four block operators is similar.

Lemma 3.2.. Let Φ be a matrix function on 𝕋 of the form (2.1) and such that ∥Φ∥_{∞} = ∥Γ_{Φ}∥. Suppose that f is a maximizing vector for Γ_{Φ} and put g = t_{0}^{−1}Γ_{Φ}f. Then Γ_{Φ}f = Φf and ∥g(ζ)∥_{ℂ^{m}} = ∥f(ζ)∥_{ℂ^{n}} a.e. on 𝕋. Furthermore ∥Φ(ζ)∥ = ∥Φ∥_{∞} a.e. on 𝕋.

Proof. We have

∥Γ_{Φ}f∥_{2} = ∥ℙ^{−}Φf∥_{2} ≤ ∥Φf∥_{2} ≤ ∥Φ∥_{∞}∥f∥_{2} = ∥Γ_{Φ}∥ · ∥f∥_{2} = ∥Γ_{Φ}f∥_{2}.

It follows that all inequalities in this chain are, in fact, equalities. The fact that ∥ℙ^{−}Φ∥_{2} = ∥Φ∥_{2} certainly means that Φf ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}) and so Γ_{Φ}f = Φf. The equality ∥Φf∥_{2} = ∥Φ∥_{∞}∥f∥_{2} implies that ∥g(ζ)∥_{ℂ^{m}} = ∥f(ζ)∥_{ℂ^{n}} a.e. on 𝕋, which in turn implies that ∥Φ(ζ)∥ = ∥Φ∥_{∞} for almost all ζ ∈ 𝕋. ■

Lemma 3.3.. Let Φ be a matrix function of the form (2.1) such that ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥. Suppose that f = f_{1}⊕f_{2} is a maximizing vector for Γ_{Φ} and t_{0}g = g_{1}⊕g_{2} = Γ_{Φ}f, where f_{1} ∈ H^{2}(ℂ^{n_{1}}), f_{2} ∈ L^{2}(ℂ^{n_{2}}), g_{1} ∈ H_{−}^{2}(ℂ^{m_{1}}), and g_{2} ∈ L^{2}(ℂ^{m_{2}}). Then

∥f_{1}(ζ)∥_{ℂ^{n_{1}}}^{2} ≥ \frac{∥Γ_{Φ}∥^{2} − ∥Γ_{Φ}∥_{e}^{2}}{∥Γ_{Φ}∥^{2}} ∥f(ζ)∥_{ℂ^{n}}^{2}, ∥g_{1}(ζ)∥_{ℂ^{m_{1}}}^{2} ≥ \frac{∥Γ_{Φ}∥^{2} − ∥Γ_{Φ}∥_{e}^{2}}{∥Γ_{Φ}∥^{2}} ∥g(ζ)∥_{ℂ^{m}}^{2}

almost everywhere on 𝕋.

Proof. By subtracting an optimal solution, we can assume that ∥Φ∥_{∞} = ∥Γ_{Φ}∥. By Lemma 3.2, t_{0}g(ζ) = Φ(ζ)f(ζ) and ∥g(ζ)∥_{ℂ^{m}} = ∥f(ζ)∥_{ℂ^{n}} a.e. on 𝕋. Therefore f(ζ) is a maximizing vector for Φ(ζ) and g(ζ) is a maximizing vector for Φ^{∗}(ζ). We have g_{2} = ( Φ_{21} Φ_{22} ) f. It is well known that

∥Γ_{( \begin{matrix}𝕆 & 𝕆 \\ Φ_{21} & Φ_{22} \end{matrix} )}∥_{e} = ∥Γ_{( \begin{matrix}𝕆 & 𝕆 \\ Φ_{21} & Φ_{22} \end{matrix} )}∥ = ∥( Φ_{21} Φ_{22} )∥_{∞}

and so

∥( Φ_{21} Φ_{22} )∥_{∞} ≤ ∥Γ_{Φ}∥_{e}.

Hence

∥Γ_{Φ}∥ · ∥g_{2}(ζ)∥_{ℂ^{m_{2}}} ≤ ∥( Φ_{21} Φ_{22} )∥_{∞} ∥f(ζ)∥_{ℂ^{n}} ≤ ∥Γ_{Φ}∥_{e}∥f(ζ)∥_{ℂ^{n}}.

On the other hand

∥g(ζ)∥_{ℂ^{m}}^{2} = ∥g_{1}(ζ)∥_{ℂ^{m_{1}}}^{2} + ∥g_{2}(ζ)∥_{ℂ^{m_{2}}}^{2} = ∥f(ζ)∥_{ℂ^{n}}^{2}.

Therefore

∥g_{1}∥_{ℂ^{m_{1}}}^{2} ≥ \frac{∥Γ_{Φ}∥^{2} − ∥Γ_{Φ}∥_{e}^{2}}{∥Γ_{Φ}∥^{2}} ∥g(ζ)∥_{ℂ^{m}}^{2}.

To prove the inequality for f_{1}(ζ) we can use the same argument since t_{0}f(ζ) = Φ^{∗}(ζ)g(ζ) and

∥( Φ_{12}^{∗} Φ_{22}^{∗} )∥ = ∥( \begin{matrix}Φ_{12} \\ Φ_{22} \end{matrix} )∥_{∞} = ∥Γ_{( \begin{matrix}𝕆 & Φ_{12} \\ 𝕆 & Φ_{22} \end{matrix} )}∥_{e} = ∥Γ_{( \begin{matrix}𝕆 & Φ_{12} \\ 𝕆 & Φ_{22} \end{matrix} )}∥ . ■

Corollary 3.4.. Let Φ be a matrix function of the form (2.1) such that ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥. Suppose that min{m_{1}, n_{1}} = 1. Then there exists a unique optimal solution of the four block problem.

Proof. It is sufficient to consider the case n_{1} = 1. To obtain the result in the case m_{1} = 1 we can pass to the transpose of Φ. Let f = f_{1} ⊕ f_{2} be a maximizing vector of Γ_{Φ} and let Q ∈ H^{∞}(M_{m_{1}1}) be an optimal solution of the four block problem, i.e.,

∥( \begin{matrix}Φ_{11} − Q & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} )∥ = ∥Γ_{Φ}∥.

It follows from Lemma 3.3 that f_{1} is a nonzero scalar function in H^{2}. By Lemma 3.2

( \begin{matrix}Φ_{11} − Q & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) = Γ_{Φ}f.

Therefore Qf_{1} is uniquely determined by Γ_{Φ} and f (Q is a column matrix) and since f_{1} ≠ 𝕆, it follows that Q is uniquely determined by Φ. ■

Now, following ideas of [PY1] we construct diagonalizing matrix functions V and W in the following way.

Take a maximizing vector f = f_{1} ⊕ f_{2}, f_{1} ∈ H^{2}(ℂ^{n_{1}}), f_{2} ∈ L^{2}(ℂ^{n_{2}}). Let h be a scalar outer function such that |h(ζ)|^{2} = ∥f(ζ)∥_{ℂ^{n}}^{2}, ζ ∈ 𝕋. Such a function h always exists since f_{1} ∈ H^{2} and f_{1} ≠ 𝕆. Denote by ϑ a greatest common inner divisor of the entries of f_{1} (it may happen that ϑ = 1). Define the vector v = v_{1} ⊕ v_{2} ∈ H^{∞}(ℂ^{n_{1}}) ⊕ L^{∞}(ℂ^{n_{2}}) by v = ϑ̄f/h. Clearly, ∥v(z)∥ = 1 a.e. on 𝕋. The vector v will be the first column of the matrix V.

We can represent the column function v_{1} as v_{1} = v_{(o)}v_{(i)} where v_{(o)} is a scalar outer function such that |v_{(o)}(ζ)| = ∥v_{1}(ζ)∥_{ℂ^{n_{1}}} a.e. on 𝕋 and v_{(i)} is an inner column function (the inner part of v_{1}), i.e., ∥v_{(i)}(ζ)∥_{ℂ^{n_{1}}} = 1 a.e. on 𝕋. Applying Lemma 3.1 to v = v_{(i)}, we obtain an inner and co-outer matrix V_{c} such that the matrix ( v_{(i)} \overline{V}_{c} ) is unitary-valued. Note that the vector-function v_{(i)} is pointwise orthogonal to any column of \overline{V}_{c} a.e. on 𝕋, and so the same is true for v_{1}. So the matrix function

( \begin{matrix}v_{1} & \overline{V}_{c} \\ v_{2} & 𝕆 \end{matrix} )  (3.1)

is isometric almost everywhere on 𝕋.

It is easy to see that we can complete this matrix function by adding m_{2} measurable column functions to obtain a unitary-valued function. Indeed it is sufficient to complete the matrix function to a square matrix function whose columns are pointwise linear independent and then apply the Gram-Schmidt orthogonalization process to the columns. To this end we can approximate our matrix function uniformly by step functions which take isometric values. Clearly, we can find a unitary completion for each step function. It is easy to see that if the distance from a step function to our initial function is sufficiently small, then the columns of our initial function and the columns we added to the step function are linearly independent.

Let V be a unitary-valued completion of the matrix (3.1). Then V has the form

V = ( \begin{matrix}v_{1} & \overline{V}_{c} & ⋆ \\ v_{2} & 𝕆 & ⋆ \end{matrix} )

Let us now construct a unitary-valued matrix W in a similar way. Let t_{0}g = Γ_{Φ}f. Then by Lemma 3.2, ∥g(ζ)∥ = ∥f(ζ)∥ = |h(ζ)|, ζ ∈ 𝕋. Let τ be a greatest common inner divisor of all entries of z̄\overline{g}_{1} (recall that g = g_{1}⊕g_{2}, g_{1} ∈ H_{−}^{2}(ℂ^{m_{1}}), g_{2} ∈ L^{2}(ℂ^{m_{2}})). Define the column function w = w_{1} ⊕ w_{2} ∈ H^{∞}(ℂ^{m_{1}}) ⊕ L^{∞}(ℂ^{m_{2}}) by w =^{def} z̄τ̄ḡ/h. By analogy with (3.1) we can find a co-outer matrix function W_{c} such that the matrix

function ( \begin{matrix}w_{1} & \overline{W}_{c} \\ w_{2} & 𝕆 \end{matrix} ) takes isometric values on 𝕋. We can complete this matrix

function to a unitary-valued matrix function and define W to be its transpose:

W^{t} = ( \begin{matrix}w_{1} & \overline{W}_{c} & ⋆ \\ w_{2} & 𝕆 & ⋆ \end{matrix} ) .

To prove Theorem 2.1 we shall proceed as follows. Let Q_{0} be an optimal solution of the four block problem. We shall prove in the next section that the matrix function

( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) admits a representation

( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 & 𝕆 \\ 𝕆 & Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ 𝕆 & Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) V^{∗},

where u_{0} = z̄ϑ̄τ̄h̄/h and Φ_{11}^{(1)} is a matrix function of size (m_{1} − 1) × (n_{1} − 1). We shall also prove in Section 4 that if Q is another optimal solution, then

( \begin{matrix}Φ_{11} − Q & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 & 𝕆 \\ 𝕆 & Φ_{11}^{(1)} − Q_{1} & Φ_{12}^{(1)} \\ 𝕆 & Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) V^{∗},

where Q_{1} ∈ H^{∞}(M_{m_{1}−1,n_{1}−1}) and

∥( \begin{matrix}Φ_{11}^{(1)} − Q_{1} & Φ_{12}^{(1)} \\ Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} )∥_{∞} ≤ t_{0}.

Since V and W are unitary-valued, it is easy to see that Q is a superoptimal solution to the four block problem for the matrix function Φ if and only if Q_{1} is a superoptimal solution to the four block problem for the matrix function

Φ^{(1)} = ( \begin{matrix}Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) .

Moreover, if t_{0}, t_{1}, ⋯, t_{d−1} is sequence of superoptimal singular values of the four block problem for Φ, then t_{1}, ⋯, t_{d−1} is the sequence of superoptimal singular values of the four block problem for Φ^{(1)}. This reduction allows us to diminish the size of the matrix function Φ_{11}.

If Γ_{Φ^{(1)}} = 𝕆, we clearly have uniqueness. To continue this process we have to be able to find a maximizing vector for the four block operator Γ_{Φ^{(1)}}. We can certainly do that if its essential norm is still less than the smallest nonzero superoptimal singular value. In Section 6 we shall prove that ∥Γ_{Φ^{(1)}}∥_{e} ≤ ∥Γ_{Φ}∥_{e} which will allow us to continue the process and reduce Theorem 2.1 to Corollary 3.4.

4. . Parametrization of optimal solutions

In this section we describe the optimal solutions of the four block problem in case when ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥.

Lemma 4.1.. Let Φ be a block matrix function of the form (2.1) such that ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥, and let V and W be the matrix functions constructed in Section 3. Then there exists a unimodular function u_{0} such that any optimal solution Q_{0} of the four block problem satisfies

( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 & 𝕆 \\ 𝕆 & Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ 𝕆 & Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) V^{∗},  (4.1)

where Φ_{11}^{(1)} is a matrix function of size m_{1} − 1 × n_{1} − 1. The unimodular function u_{0} admits a representation u_{0} = z̄b̄h̄/h, where h is an outer function in H^{2} and b is a finite Blaschke product. Moreover, the Toeplitz operator T_{u_{0}} is Fredholm and Range T_{u_{0}} = H^{2}.

Proof. Let f = f_{1} ⊕f_{2} be a maximizing vector for Γ_{Φ} and let Γ_{Φ}f = t_{0}g = t_{0}(g_{1} ⊕g_{2}). Put u_{0} = z̄ϑ̄τ̄h̄/h (see the construction of the matrix functions V and W in Section 3).

By Lemma 3.2, f(ζ) is a maximizing vector for ( \begin{matrix}Φ_{11}(ζ) − Q_{0}(ζ) & Φ_{12}(ζ) \\ Φ_{21}(ζ) & Φ_{22}(ζ) \end{matrix} ) almost

everywhere on 𝕋 and

( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) f = t_{0}g.

Therefore g(ζ) is a maximizing vector for ( \begin{matrix}Φ_{11}(ζ) − Q_{0}(ζ) & Φ_{12}(ζ) \\ Φ_{21}(ζ) & Φ_{22}(ζ) \end{matrix} )^{∗} for almost all

ζ ∈ 𝕋 and so

( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} )^{∗} g = t_{0}f.  (4.2)

Since f = ϑh ( \begin{matrix}v_{1} \\ v_{2} \end{matrix} ) and z̄ḡ = τh ( \begin{matrix}w_{1} \\ w_{2} \end{matrix} ), we have

ϑh ( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) ( \begin{matrix}v_{1} \\ v_{2} \end{matrix} ) = t_{0}z̄τ̄h̄ ( \begin{matrix}\overline{w} & 1 \\ \overline{w} & 2 \end{matrix} ) .

It follows from the definition of the matrix functions V and W (see Section 3) that

ϑh ( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) V ( 1 ; 𝕆 ; ⋮ ; 𝕆 ) = t_{0}z̄τ̄h̄W^{∗} ( 1 ; 𝕆 ; ⋮ ; 𝕆 ) .

It is easy to see that the first column of W ( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) V has the form

( t_{0}u_{0} 𝕆 ⋯ 𝕆 )^{t}, where u_{0} =^{def} z̄ϑ̄τ̄h̄/h. Similarly, using (4.2) we find that the

first row of W ( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) V has the form ( t_{0}u_{0} 𝕆 ⋯ 𝕆 ), which proves

that ( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) has the form (4.1).

Let us show that the Toeplitz operator T_{u_{0}} is Fredholm and is onto. Clearly, ∥H_{u_{0}}∥ = 1, since ∥H_{u_{0}}h∥_{2} = ∥ℙ_{−}z̄ϑ̄τ̄∥_{2} = ∥z̄ϑ̄τ̄∥_{2} = ∥h∥_{2}. We claim that ∥H_{u_{0}}∥_{e} < 1. Indeed, let f be a scalar function in H^{2}. We have

Γ_{Φ}vf = ℙ_{−} ( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) ( \begin{matrix}v_{1}f \\ v_{2}f \end{matrix} )

= ℙ_{−}W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 & 𝕆 \\ 𝕆 & Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ 𝕆 & Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) V^{∗} ( \begin{matrix}v_{1}f \\ v_{2}f \end{matrix} )

= ℙ_{−}W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 & 𝕆 \\ 𝕆 & Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ 𝕆 & Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) ( f ; 𝕆 ; ⋮ ; 𝕆 ) = t_{0}ℙ_{−} ( \begin{matrix}\overline{w}_{1}u_{0}f \\ \overline{w}_{2}u_{0}f \end{matrix} )

Therefore

ℙ_{−}w^{t}Γ_{Φ}vf = t_{0}ℙ_{−}w^{t}ℙ_{−} ( \begin{matrix}\overline{w}_{1}u_{0}f \\ \overline{w}_{2}u_{0}f \end{matrix} ) = t_{0}ℙ_{−} ( w_{1} w_{2} ) ( \begin{matrix}\overline{w}_{1}u_{0}f \\ \overline{w}_{2}u_{0}f \end{matrix} ) = t_{0}H_{u_{0}}f,

whence

t_{0}∥H_{u_{0}}∥_{e} ≤ ∥v∥_{2}∥w∥_{2}∥Γ_{Φ}∥_{e} = ∥Γ_{Φ}∥_{e} < t_{0},

which implies that ∥H_{u_{0}}∥_{e} < 1.

Since ∥H_{u_{0}}∥_{e} = dist_{L^{∞}}(u_{0}, H^{∞} + C) (see e.g., [S], [Ni]), it follows that ∥H_{u_{0}}∥_{e} = lim_{j→∞} dist_{L^{∞}}(z^{j}u_{0}, H^{∞}). We have ∥H_{u_{0}}h∥ = ∥h∥. So dist(u_{0}, H^{∞}) = ∥H_{u_{0}}∥ = 1. Therefore there exists a j ∈ ℤ_{+} such that

dist_{L^{∞}}(z^{j}u_{0}, H^{∞}) = 1 and dist_{L^{∞}}(z^{j+1}u_{0}, H^{∞}) < 1.

This means that T_{z^{j+1}u_{0}} is left invertible and T_{z^{j}u_{0}} is not left invertible which implies that T_{z^{j+1}u_{0}} is invertible (see [Ni]). Clearly, T_{z^{j+1}u_{0}} = T_{u_{0}}T_{z^{j+1}}, T_{z^{j+1}} is Fredholm and so is T_{u_{0}}.

Since u_{0} has the form u_{0} = z̄ϑ̄τ̄h̄/h, where ϑ and τ are inner and h is an outer function in H^{2}, the Toeplitz operator has dense range (see [PKh]) which together with the Fredholmness of T_{u_{0}} implies that T_{u_{0}} is onto.

It remains to show that both ϑ and τ are finite Blaschke products. Indeed, if κ is an inner divisor of ϑτ, it is easy to see that κh ∈ Ker T_{u_{0}} and since T_{u_{0}} is Fredholm, Ker T_{u_{0}} is finite dimensional, which implies that both ϑ and τ are finite Blaschke products. ■

Theorem 4.2.. Let Φ be a block matrix function of the form (2.1) such that ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥ and let Q_{0} be an optimal solution of the four block problem. Suppose that V, W, u_{0}, Φ_{11}^{(1)}, Φ_{12}^{(1)}, Φ_{21}^{(1)}, Φ_{22}^{(1)} satisfy (4.1) holds. Let Q be a matrix function of size m_{1} × n_{1}. Then Q is an optimal solution of the four block problem if and only if there exists Q_{1} ∈ H^{∞}(M_{m_{1}−1,n_{1}−1}) that satisfies the following conditions:

( \begin{matrix}Φ_{11} − Q & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 & 𝕆 \\ 𝕆 & Φ_{11}^{(1)} − Q_{1} & Φ_{12}^{(1)} \\ 𝕆 & Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) V^{∗},  (4.3)

∥( \begin{matrix}Φ_{11}^{(1)} − Q_{1} & Φ_{12}^{(1)} \\ Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} )∥_{∞} ≤ t_{0}.  (4.4)

To prove Theorem 4.2 we need the following result from [PY1]:

Lemma 4.3.. Let V, W be L^{∞} matrix functions on 𝕋, of types n × n, m × m respectively, which are unitary-valued a.e. and are of the form

V = ( v \overline{V}_{c} ) , W^{t} = ( w \overline{W}_{c} ) ,

where v, V_{c}, w, W_{c} are H^{∞} matrix functions, v and w are column functions, and V_{c}, W_{c} are co-outer. Then

WH^{∞}(M_{m,n})V ⋂ ( \begin{matrix}𝕆 & 𝕆 \\ 𝕆 & L^{∞}(M_{m−1,n−1}) \end{matrix} ) = ( \begin{matrix}𝕆 & 𝕆 \\ 𝕆 & H^{∞}(M_{m−1,n−1}) \end{matrix} ) .

Proof of Theorem 4.2. Let Q be an optimal solution. By Lemma 4.1

W ( \begin{matrix}Q_{0} − Q & 𝕆 \\ 𝕆 & 𝕆 \end{matrix} ) V

has the form

( \begin{matrix}𝕆 & 𝕆 & 𝕆 \\ 𝕆 & ⋆ & ⋆ \\ 𝕆 & ⋆ & ⋆ \end{matrix} )

(the upper left block is scalar). On the other hand it is easy to see from the definition of V and W (see Section 3) that

W ( \begin{matrix}Q_{0} − Q & 𝕆 \\ 𝕆 & 𝕆 \end{matrix} ) V = ( \begin{matrix}( w_{1} \overline{W}_{c} )^{t} (Q_{0} − Q) ( v_{1} \overline{V}_{c} ) & 𝕆 \\ 𝕆 & 𝕆 \end{matrix} ) .

Therefore

( w_{1} \overline{W}_{c} )^{t} (Q_{0} − Q) ( v_{1} \overline{V}_{c} ) = ( \begin{matrix}𝕆 & 𝕆 \\ 𝕆 & F \end{matrix} )

for some F ∈ L^{∞}(M_{m_{1}−1,n_{1}−1}) (the upper left corner of the matrix function on the right hand side is scalar). Let v_{1} = v_{(o)}v_{(i)}, w_{1} = w_{(o)}w_{(i)}, where v_{(o)} and w_{(o)} are scalar outer functions, and v_{(i)} and w_{(i)} are inner column functions. We have

( w_{(i)} \overline{W}_{c} )^{t} (Q_{0} − Q) ( v_{(i)} \overline{V}_{c} )

= ( \begin{matrix}w_{(o)} & 𝕆 \\ 𝕆 & I \end{matrix} ) ( w_{1} \overline{W}_{c} )^{t} (Q_{0} − Q) ( v_{1} \overline{V}_{c} ) ( \begin{matrix}v_{(o)} & 𝕆 \\ 𝕆 & I \end{matrix} )

= ( \begin{matrix}w_{(o)} & 𝕆 \\ 𝕆 & I \end{matrix} ) ( \begin{matrix}𝕆 & 𝕆 \\ 𝕆 & F \end{matrix} ) ( \begin{matrix}v_{(o)} & 𝕆 \\ 𝕆 & I \end{matrix} ) = ( \begin{matrix}𝕆 & 𝕆 \\ 𝕆 & F \end{matrix} ) .

Put v = v_{(i)}, w = w_{(i)}. Clearly, the matrix functions V = ( v \overline{V}_{c} ) and W =

( w \overline{W}_{c} ) satisfy the hypotheses of Lemma 4.3. Therefore F ∈ H^{∞}(M_{m_{1}−1,n_{1}−1}), which proves that Q satisfies (4.3) with Q_{1} = −F. Since Q is an optimal solution, (4.4) obviously holds.

Conversely, suppose that Q_{1} is a function in H^{∞}(M_{m_{1}−1,n_{1}−1}) satisfying (4.3). Then it follows from Lemma 4.3 that there exists a function G ∈ H^{∞}(M_{m_{1},n_{1}}) such that

( w_{1} \overline{W}_{c} )^{t} G ( v_{1} \overline{V}_{c} ) = ( \begin{matrix}𝕆 & 𝕆 \\ 𝕆 & −Q_{1} \end{matrix} ) ,

which implies that

( \begin{matrix}Φ_{11} − (G + Q_{0}) & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 & 𝕆 \\ 𝕆 & Φ_{11}^{(1)} − Q_{1} & Φ_{12}^{(1)} \\ 𝕆 & Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) V^{∗}

and so Q = G + Q_{0} ∈ H^{∞}(M_{m_{1},n_{1}}). Clearly, (4.4) implies now that Q is an optimal solution. ■

It is easy to see that Theorem 4.2 reduces the problem of finding a superoptimal

solution for Φ to the same problem for the matrix function ( \begin{matrix}Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) which has

a lower size.

5. . The matrix functions V_{c} and W_{c} are left invertible in H^{∞}

In the last section we reduced the problem of finding a superoptimal solution for

Φ to the same problem for Φ^{(1)} =^{def} ( \begin{matrix}Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ). If we could continue this process,

we would eventually reduce the problem to the case min{m_{1}, n_{1}} = 1 and it would follow from Corollary 3.4 that there is a unique superoptimal solution to the four block problem for Φ. The main problem now is to prove that the four block operator Γ_{Φ_{1}} has a maximizing vector. This is certainly the case if ∥Γ_{Φ_{1}}∥_{e} ≤ ∥Γ_{Φ}∥_{e}. To prove this inequality we use an idea of [PY2] based on the solution of the so-called matricial corona problem for the matrix functions V_{c} and W_{c}. However in our case the solvability of this corona problem is much harder than in [PY2] where V_{c}, W_{c} ∈ QC.

In this section we shall prove that the matrix functions V_{c} and W_{c} are left invertible in H^{∞} (in other words the corona problem is solvable for them) which we shall use in the next section to prove that ∥Γ_{Φ_{1}}∥_{e} ≤ ∥Γ_{Φ}∥_{e}.

Theorem 5.1.. If ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥, then the matrix functions V_{c} and W_{c} defined in Section 3 are left invertible in H^{∞}.

Clearly, it is sufficient to prove that W_{c} is left invertible in H^{∞}, which means that there exists a matrix function Ω in H^{∞}(M_{n_{1},n_{1}−1}) such that Ω(ζ)W_{c}(ζ) = I for every ζ ∈ 𝔻. To show the left invertibility of V_{c}, it is sufficient to apply Theorem 5.1 to the transposed function Φ^{t} and use the equalities ∥Γ_{Φ}∥ = ∥Γ_{Φ^{t}}∥ and ∥Γ_{Φ}∥_{e} = ∥Γ_{Φ^{t}}∥_{e}, which follow immediately from the obvious identity

Γ_{Φ^{t}} = JΓ_{Φ}^{∗}J,

where Jρ =^{def} z̄ρ̄ for a vector function ρ in L^{2}.

Recall that \overline{w} = \overline{w}_{1} ⊕ \overline{w}_{2} is the first column of W^{∗}. Denote by w_{1_{r}}, 1 ≤ r ≤ n_{1}, the components of w_{1}. We have w_{1} = w_{(o)}w_{(i)}, where w_{(o)} is a scalar outer function in H^{2} and w_{(i)} is an inner column function.

Lemma 5.2.. The vectorial Toeplitz operator T_{\overline{w}_{1}} : H^{2} → H^{2}(ℂ^{n_{1}}) is left invertible. Proof. First of all, Ker T_{\overline{w}_{1}} = \{𝕆\}. Indeed, assume that ψ ∈ Ker T_{\overline{w}_{1}}. Then \overline{w}_{1_{r}}ψ ∈ H_{−}^{2} for 1 ≤ r ≤ n_{1}. Since 1 is a greatest inner divisor of the components of w_{1}, it follows from Beurling’s theorem that the functions

{∑_{r=1}^{n_{1}} κ_{r}w_{1_{r}} : κ_{r} ∈ H^{2}}

form a dense subset in H^{2}. Therefore we can approximate ψ in the L^{1}-norm by functions of the form ∑_{r=1}^{n_{1}} κ̄_{r}w̄_{1_{r}}ψ, each of which belongs to H_{−}^{1} =^{def} {φ ∈ L^{1} : φ̂(k) = 0 for k ≥ 0}. Hence ψ ∈ H_{−}^{1} and since ψ ∈ H^{2}, it follows that ψ = 𝕆.

If T_{\overline{w}_{1}} is not left invertible, there exists a sequence of scalar functions \{φ_{j}\}_{j≥0} in H^{2} such that ∥φ_{j}∥ = 1 and φ_{n} → 𝕆 in the weak topology and ∥T_{\overline{w}_{1}}φ_{n}∥ → 0. By Lemma 4.1 the operator T_{u_{0}} is onto and so there exists a sequence {ω_{n}}_{n≥0} of scalar functions in (Ker T_{u_{0}})^{⊥} such that T_{u_{0}}ω_{n} = φ_{n}. Since T_{u_{0}} is Fredholm, ω_{n} → 𝕆 weakly. Put ρ_{n} =^{def} ω_{n}v ∈ H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}), where v is the first column of V. Let Q_{0} be an optimal solution of the four block problem for Φ. By (4.1) we have

( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) ρ_{j} = W^{∗} ( t_{0}u_{0}ω_{j} ; 𝕆 ; ⋮ ; 𝕆 ) = t_{0}u_{0}ω_{j}\overline{w} = t_{0}(φ_{j} + φ_{j}^{−})\overline{w},

for some functions φ_{j}^{−} ∈ H_{−}^{2}. It follows that

∥Γ_{Φ}ρ_{j}∥^{2} = ∥ℙ^{−} ( \begin{matrix}Φ_{11} − Q_{0} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) ρ_{j}∥^{2} = ∥t_{0}ℙ^{−}(φ_{j} + φ_{j}^{−})\overline{w}∥^{2}
= ∥t_{0}(φ_{j} + φ_{j}^{−})\overline{w}∥^{2} − ∥ℙ_{+}t_{0}(φ_{j} + φ_{j}^{−})\overline{w}_{1}∥^{2}
= ∥t_{0}u_{0}ω_{j}\overline{w}∥^{2} − ∥t_{0}ℙ_{+}φ_{j}\overline{w}_{1}∥^{2}
= ∥t_{0}u_{0}ω_{j}\overline{v}∥^{2} − ∥t_{0}ℙ_{+}φ_{j}\overline{w}_{1}∥^{2} = t_{0}^{2}(∥ρ_{j}∥^{2} − ∥T\overline{w_{1}}φ_{j}∥^{2}) ,

since ∥v(ζ)∥_{ℂ^{n_{1}}} = ∥w(ζ)∥_{ℂ^{m_{1}}}.

Taking into account that ∥T\overline{w_{1}}φ_{j}∥ → 0 and ρ_{j} → 𝕆 weakly, we obtain ∥Γ_{Φ}∥_{e} = t_{0} = ∥Γ_{Φ}∥ which contradicts the hypotheses of the lemma. ■

The next step is to prove that the Toeplitz operator T_{\overline{w}_{(i)}} is left invertible, where w_{(i)} is the inner part of w_{1}. We need the following well known facts. Let χ = {χ_{j}}_{1≤j≤k} be a column function in H^{∞}(ℂ^{k}). Then it is left invertible in H^{∞} (i.e. there exist functions κ_{j}, 1 ≤ j ≤ k, such that ∑_{j=1}^{k} κ_{j}(ζ)χ_{j}(ζ) = 1 for all ζ ∈ 𝔻) if and only if the Toeplitz operator Tχ̄ is left invertible (see [Ar]). Note that by the Carleson corona theorem (see e.g., [Ni]) χ is left invertible if and only if inf_{ζ∈𝔻} ∥χ(ζ)∥_{ℂ^{k}} > 0. This result was generalized in [SNF2] for the case of matrix (and even operator) functions: let Ξ be a matrix function in H^{∞}, then Ξ is left invertible in H^{∞} if and only if the Toeplitz operator T\overline{Ξ} is left invertible.

Lemma 5.3.. Under the hypotheses of Theorem 5.1 the Toeplitz operator T_{\overline{w}_{(i)}} is left invertible.

Proof. By Lemma 5.2, T_{\overline{w}_{1}} is left invertible. By Arveson’s theorem mentioned above w_{1} is left invertible in H^{∞}. We have w_{1} = w_{(o)}w_{(i)}, where w_{(o)} is a scalar outer function in H^{∞} and w_{(i)} is an inner column function. Obviously, it follows that w_{(i)} is left invertible in H^{∞}. Again by Arveson’s theorem this implies that T_{\overline{w}_{(i)}} is left invertible. ■

We need the following result proved in [P].

Theorem 5.4.. Let W be a unitary-valued matrix function of the form W^{t} = ( w \overline{W}_{c} ), where w is a co-outer inner column, and W_{c} is a co-outer inner function. Then the Toeplitz operator T_{W^{t}} has trivial kernel and dense range, and the operators H_{W^{t}}^{∗}H_{W^{t}} and H_{(W^{t})^{∗}}^{∗}H_{(W^{t})^{∗}} are unitarily equivalent.

The following result can easily be deduced from Theorem 5.4

Theorem 5.5.. Let W be a matrix function satisfying the hypotheses of Theorem 5.4. Suppose that ∥H\overline{w}∥ < 1. Then the Toeplitz operator T_{W^{t}} is invertible.

Proof. Clearly, ∥H_{(W^{t})^{∗}}∥ = ∥H_{w^{∗}}∥ = ∥H\overline{w}^{t}∥. It is easy to see that ∥H\overline{w}^{t}∥ = ∥H\overline{w}∥ < 1. By Theorem 5.4 the operators H_{W^{t}}^{∗}H_{W^{t}} and H_{(W^{t})^{∗}}^{∗}H_{(W^{t})^{∗}} are unitarily equivalent. Therefore ∥H_{W^{t}}∥ = ∥H_{(W^{t})^{∗}}∥ < 1. Since W^{t} takes isometric values on 𝕋, it is easy to see that

∥T_{W^{t}}F ∥_{2}^{2} + ∥H_{W^{t}}F ∥_{2}^{2} = ∥W^{t}F ∥_{2}^{2} = ∥F ∥_{2}^{2}

for every vector function F. Consequently, T_{W^{t}} is left invertible if and only if ∥H_{W^{t}}∥ < 1. It follows that both T_{W^{t}} and T_{(W^{t})^{∗}} are left invertible which means that T_{W^{t}} is invertible. ■

Proof of Theorem 5.1. Put w =^{def} w_{(i)} and let W^{t} = ( w_{(i)} \overline{W}_{c} ). By Lemma 5.3, T_{\overline{w}_{(i)}} is left invertible. Since w_{(i)} takes isometric values on 𝕋, we have as in the proof of Theorem 5.5

∥T_{\overline{w}_{(i)}}ω∥^{2} + ∥H_{\overline{w}_{(i)}}ω∥^{2} = ∥ω∥^{2}

for every ω ∈ H^{2}. Hence ∥H_{\overline{w}_{(i)}}∥ < 1 and so by Theorem 5.5 the operator T_{W^{t}} is invertible. Clearly, it follows that T_{\overline{W}_{c}} is left invertible, since T_{\overline{W}_{c}} can be interpreted as a restriction of T_{W^{t}}. Therefore by the Sz.-Nagy–Foias theorem mentioned above W_{c} is left invertible in H^{∞}. ■

6. . The essential norm of Γ_{Φ^{(1)}}

In Section 4 we reduced the proof of Theorem 2.1 to the fact that ∥Γ_{Φ_{(1)}}∥_{e} ≤ ∥Γ_{Φ}∥_{e},

where the matrix function Φ^{(1)} = ( \begin{matrix}Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) is defined in (4.1). In this section

we are going to use the facts that V_{c} and W_{c} are left invertible (see Section 5) to prove this inequality which will complete the proof of Theorem 2.1.

The idea behind the proof is the following. We use the fact that

∥Γ_{Φ^{(1)}}∥_{e} = inf{lim sup_{j} ∥Γ_{Φ^{(1)}}ξ_{j}∥_{2}},

where the infimum is taken over all sequences {ξ_{j}} in H^{2}(ℂ^{n_{1}−1}) ⊕ L^{2}(ℂ^{n_{2}}) such that ∥ξ_{j}∥_{2} = 1 and ξ_{j} → 𝕆 weakly. Given such a sequence {ξ_{j}} we construct another sequence {ρ_{j}} in H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) such that ∥ρ_{j}∥_{2} = 1, ρ_{j} → 𝕆 weakly and

lim sup_{j} ∥Γ_{Φ}ρ_{j}∥_{2} ≥ lim sup_{j} ∥Γ_{Φ^{(1)}}ξ_{j}∥_{2}.

To this end we are going to use a construction which is similar to the one used in [PY2].

Let W be the unitary-valued matrix function constructed in Section 3. Consider the matrix W^{∗} which has the form

W^{∗} = ( \begin{matrix}\overline{w}_{1} & W_{c} & F \\ \overline{w}_{2} & 𝕆 & G \end{matrix} ) .

Let

β = ( \begin{matrix}W_{c} & F \\ 𝕆 & G \end{matrix} ) .  (6.1)

To use a construction similar to the one given in [PY2], we have to find a left inverse of β of a special form. Recall that we have proved in Section 5 that W_{c} is left invertible in H^{∞}. Let W_{c}^{li} be an H^{∞} left inverse of W_{c}.

Lemma 6.1.. Let β be the matrix function defined by (6.1). Then the matrix function G is invertible in L^{∞} and there exists a bounded left inverse of β of the form

B = ( \begin{matrix}W_{c}^{li} & X \\ 𝕆 & G^{−1} \end{matrix} ) .  (6.2)

Proof. Suppose that G is not invertible in L^{∞}. Then there exists a sequence {ξ_{j}}_{j≥0} in L^{2}(ℂ^{m_{2}}) such that ∥ξ_{j}∥_{2} = 1 and ∥Gξ_{j}∥_{2} → 0.

It is easy to see from Lemma 3.3 that

∥w_{1}(ζ)∥_{ℂ^{m_{1}}}^{2} ≥ \frac{∥Γ_{Φ}∥^{2} − ∥Γ_{Φ}∥_{e}^{2}}{∥Γ_{Φ}∥^{2}} =^{def} δ < 1, ζ ∈ 𝕋.  (6.3)

Since the column \overline{w}_{1}(ζ) is orthogonal to the columns of W_{c}(ζ) a.e. on 𝕋, it follows that the matrix function ( \overline{w}_{1} W_{c} ) is invertible in L^{∞}. Therefore there exists a bounded sequence {η_{j}} in L^{2}(ℂ^{m_{1}}) such that

( \overline{w}_{1} W_{c} ) η_{j} + F ξ_{j} = 𝕆.

Then

W^{∗} ( \begin{matrix}η_{j} \\ ξ_{j} \end{matrix} ) = ( \begin{matrix}𝕆 \\ ( \overline{w}_{2} 𝕆 ) η_{j} + Gξ_{j} \end{matrix} ) .

It follows from (6.3) that ∥w_{2}(ζ)∥_{ℂ^{m_{2}}}^{2} ≤ 1 − δ, ζ ∈ 𝕋. Since ∥Gξ_{j}∥_{2} → 0, we have for large values of j

∥W^{∗} ( \begin{matrix}η_{j} \\ ξ_{j} \end{matrix} )∥_{2} < ∥η_{j}∥_{2} ≤ ∥( \begin{matrix}η_{j} \\ ξ_{j} \end{matrix} )∥_{2} ,

which contradicts the fact that W is unitary-valued.

Let now B be a matrix in the form (6.2). Clearly Bβ = I if and only if

W_{c}^{li}F + XG = 𝕆.

Since G is invertible in L^{∞}, we can always find a matrix function X in L^{∞} which satisfies this equality. ■

Remark. In the same way we can consider the submatrix α of the matrix \overline{V} constructed in Section 3,

α = ( \begin{matrix}V_{c} & ⋆ \\ 𝕆 & ⋆ \end{matrix} )

and prove that α has a left inverse in the form

A = ( \begin{matrix}V_{c}^{li} & ⋆ \\ 𝕆 & ⋆ \end{matrix} ) ,

where V_{c}^{li} is an H^{∞} left inverse of V_{c}.

To prove the main result of this section we need the following lemma which in the case of Nehari’s problem was proved in [PY2] (see Lemma 2.1 there).

Lemma 6.2.. Let η be a vector function in H_{−}^{2}(ℂ^{m_{1}−1})⊕L^{2}(ℂ^{m_{2}}) and let χ be the scalar function in H^{2} defined by

χ = −ℙ_{+}w^{t}B^{∗}η.

Then

W^{∗} ( \begin{matrix}χ \\ η \end{matrix} ) ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}).

Proof. Since W is unitary-valued, we have

I = W^{∗}W = \overline{w}w^{t} + ββ^{∗}

and hence

β = β(Bβ)^{∗} = ββ^{∗}B^{∗} = (I − \overline{w}w^{t})B^{∗}.

Therefore

W^{∗} ( \begin{matrix}χ \\ η \end{matrix} ) = ( \overline{w} β ) ( \begin{matrix}χ \\ η \end{matrix} ) = \overline{w}χ + (I − \overline{w}w^{t})B^{∗}η = B^{∗}η + \overline{w}(χ + w^{t}B^{∗}η).

It is easy to see from (6.2) that B^{∗}η ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}). Since w ∈ H^{∞}(ℂ^{m_{1}}) ⊕ L^{∞}(ℂ^{m_{2}}), it follows that \overline{w}(χ + w^{t}B^{∗}η) = \overline{w}ℙ_{−}w^{t}B^{∗}η ∈ H_{−}^{2}, which proves the result. ■

Remark. It is easy to see that if {η_{j}} is a sequence of functions in H_{−}^{2}(ℂ^{m_{1}−1}) ⊕ L^{2}(ℂ^{m_{2}}) which converges weakly to 𝕆, the above construction produces a sequence of scalar functions {χ_{j}} in H^{2}, χ_{j} = −ℙ_{+}(w^{t}B^{∗}η_{j}), which also converges weakly to 𝕆.

Now we are in a position to prove that ∥Γ_{Φ_{1}}∥_{e} ≤ ∥Γ_{Φ}∥_{e}, where the matrix function

Φ^{(1)} = ( \begin{matrix}Φ_{11}^{(1)} & Φ_{12}^{(1)} \\ Φ_{21}^{(1)} & Φ_{22}^{(1)} \end{matrix} ) is defined in (4.1).

Theorem 6.3.. Let Φ be a matrix function of the form (2.1) such that ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥. Then ∥Γ_{Φ^{(1)}}∥_{e} ≤ ∥Γ_{Φ}∥_{e}.

Proof. Let {ξ_{j}} be a sequence of functions in H_{−}^{2}(ℂ^{n_{1}−1})⊕L^{2}(ℂ^{n_{2}}) such that ∥ξ_{j}∥_{2} = 1 and ξ_{j} → 𝕆 weakly. Put η_{j} = Γ_{Φ^{(1)}}ξ_{j}. We are going to construct a sequence of

functions \{ξ_{j}^{#}\} in H_{−}^{2}(ℂ^{n_{1}})⊕L^{2}(ℂ^{n_{2}}) such that \frac{ξ_{j}^{#}}{∥ξ_{j}^{#}∥_{2}} → 𝕆 weakly, and \frac{∥η_{j}^{#}∥_{2}}{∥ξ_{j}^{#}∥_{2}} ≥ ∥η_{j}∥_{2},

where η_{j}^{#} =^{def} Γ_{Φ}ξ_{j}^{#}. As we have explained in the beginning of the section, this would

imply the desired inequality (put ρ_{j} = \frac{ξ_{j}^{#}}{∥ξ_{j}^{#}∥_{2}} ).

To this end we apply Lemma 6.2 to the sequence {η_{j}}. We obtain a sequence of scalar H^{2} functions {χ_{j}} such that

W^{∗} ( \begin{matrix}χ_{j} \\ η_{j} \end{matrix} ) ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}).

Put

ξ_{j}^{#} = A^{t}ξ_{j} + q_{j}v,

where v is the first column of V and A is the left inverse of α described in the Remark after Lemma 6.1. The scalar functions q_{j} will be chosen later.

We have

( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) V^{∗}ξ_{j}^{#} = ( \begin{matrix}t_{0}u_{0}q_{j} + t_{0}u_{0}v^{∗}A^{t}ξ_{j} \\ Φ^{(1)}ξ_{j} \end{matrix} ) .  (6.4)

Since the Toeplitz operator T_{u_{0}} is onto, we can pick q_{j} as a solution of the equation

ℙ_{+}(t_{0}u_{0}q_{j} + t_{o}u_{o}v^{∗}A^{t}ξ_{j}) = χ_{j}.

Clearly, we may choose the q_{j} so that q_{j} → 𝕆 weakly. Indeed, we may put

q_{j} = t_{0}^{−1}(T_{u_{0}}|(Ker T_{u_{0}})^{⊥})^{−1}(t_{0}χ_{j} − ℙ_{+}u_{0}v^{∗}A^{t}ξ_{j}).

It follows that ξ_{j}^{#} → 𝕆 weakly.

Let us show that the sequence {ξ_{j}^{#}} has the required properties. Since ξ_{j}^{#} → 𝕆

weakly, to prove that \frac{ξ_{j}^{#}}{∥ξ_{j}^{#}∥_{2}} → 𝕆 weakly, we have to estimate ∥ξ_{j}^{#}∥_{2} from below. We have

∥ξ_{j}^{#}∥_{2}^{2} = ∥V^{∗}ξ_{j}^{#}∥_{2}^{2} = ∥q_{j} + v^{∗}A^{t}ξ_{j}∥_{2}^{2} + ∥ξ_{j}∥_{2}^{2} ≥ 1.  (6.5)

To complete the proof it remains to show that ∥Γ_{Φ}ξ_{j}^{#}∥_{2} ≥ ∥η_{j}∥_{2}.

Recall that η_{j} = Γ_{Φ^{(1)}}η_{j} = ℙ^{−}Φ^{(1)}ξ_{j} and so Φ^{(1)}ξ_{j} − η_{j} ∈ H^{2}(ℂ^{m_{1}−1}) ⊕ {𝕆}. It is easy to see from the definition of W (see Section 3) that

W^{∗} ( \begin{matrix}𝕆 \\ Φ^{(1)}ξ_{j} − η_{j} \end{matrix} ) ∈ H^{2}(ℂ^{m_{1}}) ⊕ \{𝕆\}.

It follows now from (6.4) that

η_{j}^{#} = ℙ^{−}Φξ_{j}^{#} = ℙ^{−}W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) V^{∗}ξ_{j}^{#} = ℙ^{−}W^{∗} ( \begin{matrix}χ_{j} + ω_{j} \\ η_{j} \end{matrix} ) ,

where

ω_{j} =^{def} ℙ_{−}(t_{0}u_{0}q_{j} + t_{0}u_{0}v^{∗}A^{t}ξ_{j}).

Since the first column of W^{∗} is \overline{w}_{1} ⊕ \overline{w}_{2} and w_{1} ∈ H^{∞}, it follows that

W^{∗} ( \begin{matrix}ω_{j} \\ 𝕆 \end{matrix} ) ∈ ( \begin{matrix}H_{−}^{2}(ℂ^{m_{1}}) \\ L^{2}(ℂ^{m_{2}}) \end{matrix} ) .

We have chosen {χ_{j}} so that

W^{∗} ( \begin{matrix}χ_{j} \\ η_{j} \end{matrix} ) ∈ ( \begin{matrix}H_{−}^{2}(ℂ^{m_{1}}) \\ L^{2}(ℂ^{m_{2}}) \end{matrix} ) .

Therefore

ℙ^{−}W^{∗} ( \begin{matrix}χ_{j} + ω_{j} \\ η_{j} \end{matrix} ) = W^{∗} ( \begin{matrix}χ_{j} + ω_{j} \\ η_{j} \end{matrix} ) .

Hence

∥η_{j}^{#}∥_{2}^{2} = ∥χ_{j} + ω_{j}∥_{2}^{2} + ∥η_{j}∥_{2}^{2} = t_{0}^{2}∥q_{j} + v^{∗}A^{t}ξ_{j}∥_{2}^{2} + ∥Γ_{Φ^{(1)}}ξ_{j}∥^{2}.

Since ∥Γ_{Φ^{(1)}}∥ ≤ t_{0}, this together with (6.5) yields

\frac{∥η_{j}^{#}∥_{2}}{∥ξ_{j}^{#}∥_{2}} ≥ \frac{∥η_{j}∥_{2}}{∥ξ_{j}∥_{2}} = ∥η_{j}∥_{2},

which completes the proof. ■

As we have already explained, Theorem 6.3 allows us to complete the proof of Theorem 2.1.

Proof of Theorem 2.1. By Theorem 4.2 the four block for Φ has a unique solution if so does the four block problem for Φ^{(1)} and the superoptimal singular values of the four block problem for Φ^{(1)} are t_{1}, t_{2}, ⋯, t_{d−1}. By Theorem 6.3, ∥Γ_{Φ^{(1)}}∥_{e} ≤ ∥Γ_{Φ}∥_{e}. If Γ_{Φ^{(1)}} = 𝕆, we certainly have uniqeness. Otherwise we can continue this process.

Doing in this way we may stop the process if we get on a certain stage the zero four block operator or, otherwise, we eventually reduce the problem to the case d = 1. Uniqueness follows now from Corollary 3.4.

The fact that the singular values are constant on 𝕋 follows immediately from the facts that V and W are unitary-valued and u_{0} is unimodular, and from Lemma 3.2. ■

7. . Thematic factorizations and indices of superoptimal singular values

In this section we analyze the algorithm described in Section 3 and obtain certain special factorizations of superoptimal symbols of four block operators satisfying the hypotheses of Theorem 2.1. Following [PY1] we shall call such factorizations thematic.

In Section 3 we have constructed matrix functions V and W associated with the four block problem. By analogy with [PY1] we shall call matrix functions of the form V or W thematic functions.

To state the result we may assume without loss of generality that n_{1} ≤ m_{1} (otherwise we can take the transpose).

Theorem 7.1.. Let Φ be a superoptimal symbol of the four block operator Γ which satisfies the hypotheses of Theorem 2.1 and suppose that n_{1} ≤ m_{1}. Then Φ admits the following factorization

Φ = W_{0}^{∗}W_{1}^{∗}W_{2}^{∗} ⋯ W_{d−1}^{∗}DV_{d−1}^{∗} ⋯ V_{2}^{∗}V_{1}^{∗}V_{0}^{∗}  (7.1)

where

D = ( \begin{matrix}t_{0}u_{0} & 𝕆 & ⋯ & 𝕆 \\ 𝕆 & t_{1}u_{1} & ⋯ & 𝕆 \\ ⋮ & ⋮ & ... & ⋮ \\ 𝕆 & 𝕆 & ⋯ & t_{d−1}u_{d−1} & 𝕆 \\ 𝕆 & 𝕆 & ⋯ & 𝕆 \\ ⋮ & ⋮ & ... & ⋮ \\ 𝕆 & 𝕆 & ⋯ & 𝕆 \\ & & 𝕆 & & ⋆ \end{matrix} ) ,

the u_{j} are unimodular functions such that the Toeplitz operator T_{u_{j}} is Fredholm and ind T_{u_{j}} > 0, and the matrix functions W_{j} and V_{j} have the form

V_{j} = ( \begin{matrix}I_{j} & 0 \\ 0 & V̆_{j} \end{matrix} ) W_{j} = ( \begin{matrix}I_{j} & 0 \\ 0 & W̆_{j} \end{matrix} ) ,

where V̆_{j}, W̆_{j} are thematic matrix functions and I_{j} is the identity j × j matrix.

It is easy to see that the successive application of the algorithm described in Section 3 gives us a desired factorization.

Remark. As in the case of Nehari’s problem (see [PY1]) it is easy to see that if a matrix function admits a factorization of the form (7.1), then it is the superoptimal symbol of the corresponding four block operator.

We can associate with the factorization (7.1) the factorization indices k_{j} which are defined in the case t_{j} ≠ 0. We put k_{j} = ind T_{u_{j}} = dim Ker T_{u_{j}}.

It was shown in [PY1] that even for Nehari’s problem the indices depend on the choice of a thematic factorization rather than on the function Φ itself. However, it was shown in [PY2] that for Nehari’s problem with compact Hankel operator the sum of the indices corresponding to equal superoptimal singular values is an invariant (i.e. does not depend on the choice of a factorization).

The same turns out to be true for the four block problem too, and we shall prove this later in Section 9. Moreover, the sum of the indices corresponding to equal superoptimal singular values admits a quite natural and simple geometric interpretation. To give this interpretation we have to introduce a new object — the so–called superoptimal weight.

8. . Superoptimal weight

Let W ∈ L^{∞}(M_{n,n}) be a matrix weight, i.e. a bounded matrix-valued function on 𝕋, whose values are nonnegative selfadjoint n × n matrices.

Given a four block operator Γ : H^{2}(ℂ^{n_{1}})⊕L^{2}(ℂ^{n_{2}}) → H_{−}^{2}(ℂ^{m_{1}})⊕L^{2}(ℂ^{m_{2}}), n_{1}+n_{2} = n, we call a weight W admissible if

∥Γf∥^{2} ≤ (Wf, f) =^{def} ∫_{𝕋}(W(ζ)f(ζ), f(ζ))dm(ζ), f ∈ H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}).

We need the following result which we call Generalized Nehari’s Theorem.

Theorem 8.1.. Given a four block operator Γ_{Φ} and an admissible weight W there exists a symbol Φ of Γ (i.e. an operator-valued function Φ such that Γ = Γ_{Φ}) satisfying Φ^{∗}Φ ≤ W.

If Φ is a symbol of Γ satisfying Φ^{∗}Φ ≤ W, we say that Φ is dominated by the admissible weight W.

In the case W ≡ cI, v ∈ ℝ_{+}, this result was established in [FT], and this is an analog of Nehari’s theorem for four block operators. In the general case the result follows from Theorem 1.1 of [TV], since the four block operator Γ acting from the space H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) endowed with the weighted norm ∥ · ∥_{W} to the space H^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}) satisfies the hypothesis of the theorem.

For the sake of completeness we deduce Theorem 8.1 from the analog of Nehari’s theorem mentioned above.

Proof of theorem 8.1. Define W_{ε} = W + εI. Since W_{ε} ≥ εI, it admits a factorization W_{ε} = G_{ε}^{∗}G_{ε}, where G_{ε} ∈ H^{∞}(M_{n,n}) is a matrix function which is invertible in H^{∞} (see [R]). The weight W_{ε} is clearly admissible, so

∥Γf∥ ≤ ∥G_{ε}f∥ , f ∈ H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) ,

which is equivalent to the fact that

∥ΓG_{ε}^{−1}f∥ ≤ ∥f∥ , f ∈ H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) ,

Since G_{ε}^{−1} ∈ H^{∞}(M_{n,n}), we can consider the operator ΓG_{ε}^{−1} as a four block operator. By the analog of Nehari’s theorem it has a symbol Ψ_{ε} such that ∥Ψ_{ε}∥_{∞} ≤ 1. Then the function Φ_{ε} = ΨG_{ε} is a symbol of Γ and

Φ_{ε}^{∗}Φ_{ε} = G_{ε}^{∗}Ψ_{ε}^{∗}Ψ_{ε}G_{ε} ≤ G_{ε}^{∗}G_{ε} = W_{ε} = W + εI.

It remains to chose a sequence {ε_{j}} converging to 0 and such that the sequence {Φ_{ε_{j}}} converges to a matrix function, say Φ ∈ L^{∞}, in the ∗-weak topology. Clearly, Φ is a symbol of Γ dominated by W. ■

Definition. Let W be an admissible weight for the four block operator G. Consider the numbers

s_{j}^{∞}(W) =^{def} ess sup_{ζ∈𝕋}s_{j}(W(ζ)), 0 ≤ j ≤ d − 1, d = min{m_{1}, n_{1}}.

The admissible weight W is called superoptimal if it lexicographically minimizes the numbers s_{0}^{∞}(W), s_{1}^{∞}(W), ⋯, s_{d−1}^{∞}(W) among all admissible weights, i.e.,

s_{0}^{∞}(W) = min{s_{0}^{∞}(V) : V is admissible},

s_{1}^{∞}(W) = min{s_{1}^{∞}(V) : V is admissible, s_{0}^{∞}(V) is minimal possible}, etc.

The following lemma shows that under the the hypotheses of Theorem 2.1 a superoptimal weight exists. However, a superoptimal weight is not unique in general. The lemma also shows that a superoptimal weight is nevertheless “essentially” unique for our purposes.

Let λ_{a}, a ∈ ℝ, be the function on ℝ defined by

λ_{a}(t) =^{def} \{ \begin{matrix}t, & t ≥ a \\ 0, & t < a \end{matrix} .

Lemma 8.2.. Let Γ be a four block operator satisfying the hypothesis of Theorem 3.3. Let Φ be the superoptimal symbol of Γ. Then

1. Φ^{∗}Φ is a superoptimal weight for Γ;

2. If W and W′ are two superoptimal weights, then λ_{a}(W) = λ_{a}(W′) for any a ≥ t_{d−1}.

Proof. Note that Φ is a symbol of Γ dominated by the weight Φ^{∗}Φ. Suppose that Φ^{∗}Φ is not a superoptimal weight, i.e. that there exists an admissible weight W such that for some j_{0}, 0 ≤ j_{0} ≤ d − 1,

s_{j_{0}}^{∞}(W) < s_{j_{0}}^{∞}(Φ^{∗}Φ), s_{j}^{∞}(W) = s_{j}^{∞}(Φ^{∗}Φ), 0 ≤ j ≤ j_{0}.

Let Ψ be a symbol of Γ dominated by the weight W. Then

s_{j_{0}}^{∞}(Ψ) < s_{j_{0}}^{∞}(Φ), s_{j}^{∞}(Ψ) = s_{j}^{∞}(Φ), 0 ≤ j ≤ j_{0},

which contradicts the fact that Φ is the superoptimal symbol of Γ. Therefore Φ^{∗}Φ is a superoptimal weight.

Let now W be a superoptimal weight, and let Ψ be a symbol of Γ dominated by W. Then Ψ lexicographically minimizes (s_{0}^{∞}(Ψ), s_{1}^{∞}(Ψ), ...s_{d−1}^{∞}(Ψ)) and so Ψ coincides with the superoptimal symbol Φ. So, for any superoptimal weight W the superoptimal symbol Φ is the unique symbol of Γ dominated by W. This means that Φ^{∗}Φ ≤ W for any superoptimal weight W. Together with the equalities s_{j}^{∞}(W) = s_{j}^{∞}(Φ^{∗}Φ) = t_{j}^{2} this implies the second part of the lemma. ■

Denote by Λ_{a} the function defined by

Λ_{a}(t) =^{def} \{ \begin{matrix}t, & t ≥ a \\ a, & t < a \end{matrix}  (8.1)

The following fact is an easy consequence of Lemma 8.2.

Corollary 8.3.. Let W and W′ be two superoptimal weights. Then Λ_{a}(W) = Λ_{a}(W′) for any a ≥ t_{d−1}.

It is easy to see that if a = t_{d−1} and W is a superoptimal weight, the weight Λ_{a}(W) is the (unique) maximal superoptimal weight.

9. . Invariance of indices

The main result of this section shows that the sum of the indices of a thematic factorization of a superoptimal symbol does not depend on the choice of a factorization. To prove this fact we shall use the same construction which was used in Section 6 to prove Theorem 6.3.

Let

a_{0} > a_{1} > ⋯ > a_{l}

be all the distinct nonzero superoptimal singular values of a four block operator Γ which satisfies the hypotheses of Theorem 2.1. Let Φ be the superoptimal symbol of Γ and let k_{j} be the indices of a thematic factorization of Φ of the form (7.1). Consider the sum of the indices that correspond to equal superoptimal singular values:

ν_{r} = ∑_{{j:t_{j}=a_{r}}} k_{j}, 0 ≤ r ≤ l.

The following theorem is the main result of the section.

Theorem 9.1.. The numbers ν_{r} do not depend on the choice of thematic factorization of Φ.

We are going to deduce Theorem 9.1 from Theorem 9.3 below, which describes the numbers ν_{r} in terms of a superoptimal weight W.

We say that a nonzero function ξ ∈ H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) is a maximizing vector for an admissible weight W if

∥Γξ∥^{2} = (Wξ, ξ).

Lemma 9.2.. Let Γ be a four block operator, W an admissible weight for Γ, and Φ a symbol of Γ dominated by W. Let ξ be a maximizing vector for W. Then Φξ ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}), i.e., Γ_{Φ}ξ = Φξ.

Note that for W ≡ cI, c ∈ ℝ_{+}, this was proved in Lemma 3.2.

Proof. We have

(Wξ, ξ) = ∥Γ_{Φ}ξ∥_{2}^{2} = ∥ℙ^{−}Φξ∥_{2}^{2} ≤ ∥Φξ_{2}∥^{2} ≤ (Wξ, ξ) .

It follows that ∥ℙ^{−}Φξ∥_{2}^{2} ≤ ∥Φξ_{2}∥^{2}, which implies the result. ■

Given an admissible weight W put

E(W) = {ξ ∈ H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) : ξ is maximizing for W or ξ = 𝕆}.

It is easy to see that ξ ∈ E(W) if and only if ℙ^{+}Wξ − Γ_{Φ}^{∗}Γ_{Φ}ξ = 𝕆, where ℙ^{+} is the orthogonal projection onto H^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}). Therefore E(W) is a closed linear subspace. Recall that Λ_{a}(W) does not depend on the choice of superoptimal weight, where Λ_{a} is defined in (8.1).

Theorem 9.3.. Let Γ be a four block operator satisfying the hypotheses of Theorem 2.1, W a superoptimal admissible weight, and Φ the superoptimal symbol of Γ. Consider a thematic factorization of Φ of the form (7.1). Let k_{j} be the indices of the factorization. Then for a ≥ a_{l},

∑_{{j:t_{j}≥a}} k_{j} = dim E(Λ_{a}(W)) .  (9.1)

Let us first deduce Theorem 9.1 from Theorem 9.3.

Proof of theorem 9.1. It follows immediately from (9.1) that

ν_{0} = dim E(Λ_{a_{0}}(W)), ν_{j} = dim E(Λ_{a_{j}}(W)) ⊖ E(Λ_{a_{j−1}}(W)), 1 ≤ j ≤ l,

which proves the result. ■

Proof of Theorem 9.3. It is easy to see that E(Λ_{a}(W)) is constant on (a_{j+1}, a_{j}]. So it is sufficient to prove that for 0 ≤ s ≤ l

∑_{{j:t_{j}≥a_{s}}} k_{j} = dim E(W_{s}) ,

where W_{s} = Λ_{a_{s}}(W).

Let us prove the theorem by induction on d.

If d = 1, factorization (7.1) has the form

Φ = W_{0}^{∗}DV_{0}^{∗} ,

where

D = ( \begin{matrix}t_{0}u_{0} \\ 𝕆 \\ ⋮ & 𝕆 \\ 𝕆 \\ 𝕆 & ⋆ \end{matrix} ) ,

and ∥⋆∥ < t_{0} (otherwise the essential norm of Γ_{Φ} would not be less than t_{0}). Clearly, W_{0}(ζ) ≡ a_{0}I. If ξ is a maximizing vector for W_{0}, then it is easy to see that only the first entry of V_{0}^{∗}ξ is nonzero. Therefore ξ(ζ) is pointwise orthogonal to all columns of V except for the first one. It follows that ξ = χv, where χ is a scalar function in L^{2}. Using the fact that v_{1} is a co-outer column function, one can easily deduce that χ ∈ H^{2}. It is easy to see that

Φξ = t_{0}u_{0}χ\overline{w}.

Since ξ is a maximizing vector, it follows that Φξ ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}). We can now use the fact that w_{1} is a co-outer column function to deduce that u_{0}χ ∈ H_{−}^{2} which means that χ ∈ Ker T_{u_{0}}.

Conversely, it is easy to see that if χ ∈ Ker T_{u_{0}}, then ξ = χv is a maximizing vector, which proves that dim E(W_{0}) = dim Ker T_{u_{0}} = k_{0}.

Suppose now that the theorem is proved for d − 1. We have

Φ = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) V^{∗},

where Φ^{(1)} is the superoptimal symbol of Γ_{Φ^{(1)}}, W =^{def} W_{0}, and V =^{def} V_{0}. The induction hypothesis implies that the theorem holds for Γ_{Φ^{(1)}}.

Let 0 ≤ s ≤ l and a = a_{s}. Suppose that W′ is a superoptimal weight for Γ_{Φ^{(1)}}. By the induction hypothesis

dim E(Λ_{a}(W′)) = N =^{def} ∑_{{j≥1:t_{j}≥a}} k_{j},

where the k_{j} are the indices of the thematic factorization (7.1). By Lemma 9.2, ξ ∈ E(Λ_{a}(W′)) if and only if

Φ^{(1)}ξ ∈ H_{−}^{2}(ℂ^{m_{1}−1}) ⊕ L^{2}(ℂ^{m_{2}}) and ∥Φ^{(1)}ξ∥^{2} = (Λ_{a}(W′)ξ, ξ).

Let ξ_{1}, ξ_{2}, ⋯, ξ_{N} be a basis in E(Λ_{a}(W′)) and let η_{ι} = Γ_{Φ^{(1)}}ξ_{ι}. By Lemma 6.2 there exist scalar functions χ_{ι}, 1 ≤ ι ≤ N, such that

W^{∗} ( \begin{matrix}χ_{ι} \\ η_{ι} \end{matrix} ) ∈ H_{−}^{2}(ℂ^{m_{1}−1}) ⊕ L^{2}(ℂ^{m_{2}}).

As in the proof of Theorem 6.3 we define the functions ξ_{ι}^{#} as

ξ_{ι}^{#} = A^{t}ξ_{ι} + q_{ι}v,

where q_{ι} is a scalar functions in H^{2} satisfying

ℙ_{+}(t_{0}u_{0}q_{ι} + t_{0}u_{0}v^{∗}A^{t}ξ_{ι}) = χ_{ι}

(recall that the matrix function A is defined in after Lemma 6.1). We have

η_{ι}^{#} =^{def} ℙ^{−}Φξ_{ι}^{#} = ℙ^{−}W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) V^{∗}ξ_{ι}^{#} = ℙ^{−}W^{∗} ( \begin{matrix}χ_{ι} + ω_{ι} \\ η_{ι} \end{matrix} ) ,

where as in the proof of Theorem 6.3

ω_{ι} = ℙ_{−}(t_{0}u_{0}q_{ι} + t_{0}u_{0}v^{∗}A^{t}ξ_{ι}).

As we have explained in the proof of Theorem 6.3

ℙ^{−}W^{∗} ( \begin{matrix}χ_{ι} + ω_{ι} \\ η_{ι} \end{matrix} ) = W^{∗} ( \begin{matrix}χ_{ι} + ω_{ι} \\ η_{ι} \end{matrix} )

and so η_{ι}^{#} = Φξ_{ι}^{#}.

Since the matrix function W is unitary-valued, we have

∥Φξ_{ι}^{#}∥^{2} = ∥W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) ( \begin{matrix}q_{ι} + v^{∗}A^{t}ξ_{ι} \\ ξ_{ι} \end{matrix} )∥^{2}
= t_{0}^{2}∥u_{0}(q_{ι} + v^{∗}A^{t}ξ_{ι})∥^{2} + ∥Φ^{(1)}ξ_{ι}∥^{2}
= t_{0}^{2}∥q_{ι} + v^{∗}A^{t}ξ_{ι}∥^{2} + (Λ_{a}(W′)ξ_{ι}, ξ_{ι})

(the last equality holds because ξ_{ι} ∈ E(Λ_{a}(W′)), where W′ = (Φ^{(1)})^{∗}Φ^{(1)} is a superoptimal weight for Γ_{Φ^{(1)}}).

Consider the weight V,

V = ( \begin{matrix}t_{0}^{2} & 𝕆 \\ 𝕆 & W′ \end{matrix} ) .

Bearing in mind that

V^{∗}ξ_{ι}^{#} = ( \begin{matrix}q_{ι} + v^{∗}A^{t}ξ_{ι} \\ ξ_{ι} \end{matrix} ) ,

we can continue the above chain of inequalities:

∥Φξ_{ι}^{#}∥^{2} = (Λ_{a}(V)V^{∗}ξ_{ι}^{#}, V^{∗}ξ_{ι}^{#}) = (Λ_{a}(W)ξ_{ι}^{#}, ξ_{ι}^{#})  (9.2)

(the last equality holds because V is unitary-valued). Since Γ_{Φ}ξ_{ι}^{#} = ℙ^{−}Φξ_{ι}^{#} = Φξ_{ι}^{#}, it follows from (9.2) that ξ_{ι}^{#} ∈ E(Λ_{a}(W)).

We can add now another k_{0} linear independent vectors of E(Λ_{a}(W)). Let x_{1}, ⋯, x_{k_{0}} be a basis of Ker T_{u_{0}}. Obviously, x_{ι}v ∈ E(Λ_{a}(W)). Let us show that the vectors ξ_{1}^{#}, ⋯, ξ_{N}^{#}, x_{1}v, ⋯, x_{k_{0}}v are linearly independent. It is sufficient to prove that if x ∈ Ker T_{u_{0}} and xv + ∑_{ι=1}^{N} c_{ι}ξ_{ι}^{#} = 𝕆, then x = 𝕆 and c_{ι} = 0, 1 ≤ ι ≤ N. We have

V^{∗}(xv + ∑_{ι=1}^{N} c_{ι}ξ_{ι}^{#}) = ( \begin{matrix}x \\ 𝕆 \end{matrix} ) + ∑_{ι=1}^{N} c_{ι} ( \begin{matrix}v^{∗}A^{t}ξ_{ι} + q_{ι} \\ ξ_{ι} \end{matrix} ) = 𝕆.  (9.3)

Since the ξ_{ι} are linearly independent, it follows that c_{ι} = 0, 1 ≤ ι ≤ N, which in turn implies that x = 𝕆.

This proves that

∑_{{j:t_{j}≥a}} k_{j} ≤ dim E(Λ_{a}(W)).

Let us prove the opposite inequality.

Denote by E_{0} the set of vectors in E(Λ_{a}(W)) of the form xv such that x is a scalar function in H^{2}. It is easy to see that xv ∈ E_{0} if and only if x ∈ Ker T_{u_{0}}. It remains to show that there exists at most ∑_{{j>0:t_{j}≥a}} k_{j} vectors ξ̆_{ι} that are linearly independent

modulo E_{0}. Let η̆_{ι} =^{def} Γ_{Φ}ξ̆_{ι}. By Lemma 9.2, η̆_{ι} = Φξ̆_{ι}. Put

V^{∗}ξ̆_{ι} = ( \begin{matrix}γ_{j} \\ ξ_{ι} \end{matrix} ) , W^{∗}η̆_{ι} = ( \begin{matrix}δ_{ι} \\ η_{ι} \end{matrix} ) ,

where γ_{ι}, δ_{ι} are scalar functions in L^{2}. Since the vectors ξ̆_{ι} are linearly independent modulo E_{0}, the vectors ξ_{ι} are linearly independent. To complete the proof, it is sufficient to show that ξ_{ι} ∈ E(Λ_{a}(W′)).

Since η̆_{ι} = Φξ̆_{ι}, we have that η_{ι} = Φ^{(1)}ξ_{ι} and δ_{ι} = t_{0}u_{0}γ_{ι}. It follows from the block structure of V and W that ξ_{ι} ∈ H^{2}(ℂ^{n_{1}−1})⊕L^{2}(ℂ^{n_{2}}) and η_{ι} ∈ H_{−}^{2}(ℂ^{m_{1}−1})⊕L^{2}(ℂ^{m_{2}}). So η_{ι} = Φ^{(1)}ξ_{ι} = Γ_{Φ^{(1)}}ξ_{ι}.

To show that ξ_{ι} ∈ E(Λ_{a}(W′)), consider the following chain of equalities

(Λ_{a}(W)ξ̆_{ι}, ξ̆_{ι}) = (Λ_{a}(V WV^{∗})V ξ̆_{ι}, V ξ̆_{ι})
= (( \begin{matrix}t_{0}^{2} & 𝕆 \\ 𝕆 & Λ_{a}(W′) \end{matrix} ) ( \begin{matrix}γ_{ι} \\ ξ_{ι} \end{matrix} ) , ( \begin{matrix}γ_{ι} \\ ξ_{r} \end{matrix} ))
= t_{0}^{2}∥γ_{ι}∥^{2} + (Λ_{a}(W′)ξ_{ι}, ξ_{ι}) .

On the other hand

(Λ_{a}(W)ξ̆_{ι}, ξ̆_{ι}) = ∥Φξ̆_{ι}∥ = ∥η̆_{ι}∥ = ∥η_{ι}∥^{2} + ∥δ_{ι}∥^{2} = ∥Φ^{(1)}ξ_{ι}∥^{2} + t_{0}^{2}∥γ_{ι}∥^{2}.

Therefore (Λ_{a}(W′)ξ_{ι}, ξ_{ι}) = ∥Φ^{(1)}ξ_{ι}∥^{2}, which implies ξ_{ι} ∈ Λ_{a}(W′). ■

10. . Singular values of Γ_{Φ} and superoptimal singular values

Let Γ be a four block operator satisfying the hypotheses of Theorem 2.1. Denote by Φ its unique superoptimal symbol and consider a thematic factorization of Φ of the form (7.1). Let {t_{j}} be the superoptimal singular values and {k_{j}} the indices of the factorization. Consider the extended t-sequence for Γ:

t_{0}, t_{0}, ⋯, t_{1}, t_{1}, ⋯, t_{1}, ⋯

in which t_{j} is repeated k_{j} times. We denote the terms of the extended sequence by t′_{0}, t′_{1}, t′_{2}, · · ·. Although the indices k_{j} depend on the choice of thematic factorization, it follows from Theorem 9.1 that the extended t-sequence is uniquely determined by Γ.

In [PY2] it was shown in the case of Nehari’s problem with Φ ∈ H^{∞} + C that t′_{j} ≤ s_{j}(H_{Φ}), 0 ≤ j ≤ k − 1. In this section we are going to prove the same inequality in the case of the four block problem under the hypotheses of Theorem 2.1. Moreover, we prove in this section a stronger result which is also new in the case of Nehari’s problem with an H^{∞} + C symbol. To prove the results we use in this section the same machinery as we used in Section 6.

Let Γ be a four block operator that satisfies the hypotheses of Theorem 2.1. Then (see Section 3)

Φ = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) V^{∗},

where the unitary-valued matrix functions V and W are defined in Section 3. The following inequality is the main result of the section.

Theorem 10.1.. Let Γ be a four block operator such that ∥Γ_{Φ}∥_{e} < ∥Γ_{Φ}∥ and let Φ be its superoptimal symbol. Then

s_{j}(Γ_{Φ^{(1)}}) ≤ s_{j+k_{0}}(Γ_{Φ}), j ∈ ℤ_{+}.

Recall that k_{0} = dim Ker T_{u_{0}}.

Let us first derive from Theorem 10.1 the desired inequality between the singular values of Γ_{Φ} and the superoptimal singular values.

Theorem 10.2.. Under the hypotheses of Theorem 2.1

t′_{j} ≤ s_{j}(Γ_{Φ}), j ≥ 0.

Proof of Theorem 10.2. Let x ∈ Ker T_{u_{0}}. Clearly,

Γ_{Φ}xv = W^{∗} ( t_{0}u_{0}x 𝕆 ⋯ 𝕆 )^{t} = t_{0}u_{0}x\overline{w} ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}).

It follows that ∥Γ_{Φ}xv∥_{2} = t_{0}∥xv∥_{2} which proves that

s_{j}(Γ_{Φ}) = t_{0}, 0 ≤ j ≤ k_{0} − 1.  (10.1)

We can now proceed by induction on d. Clearly, the result holds for d = 1. It is also obvious that if the theorem holds for Γ_{Φ^{(1)}}, then by Theorem 10.1

t′_{j} ≤ s_{j}(Γ_{Φ}), k_{0} ≤ j ≤ d − 1,

which together with (10.1) proves the theorem. ■

Proof of Theorem 10.1. Clearly, it is sufficient to prove the following fact. Let L be a subspace of H_{−}^{2}(ℂ^{n_{1}−1})⊕L^{2}(ℂ^{n_{2}}) such that ∥Γ_{Φ^{(1)}}ξ∥_{2} ≥ s∥ξ∥_{2}, for every ξ ∈ L, where 0 < s ≤ t_{0}, then there exists a subspace M of H_{−}^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) such that dim M ≥ dim L + k_{0} and ∥Γ_{Φ}ρ∥_{2} ≥ s∥ρ∥_{2} for every ρ ∈ M.

Let ξ_{ι}, 1 ≤ ι ≤ N, be a basis in L. Put η_{ι} = Γ_{Φ^{(1)}}ξ_{ι}. By Lemma 6.2 there exist

scalar functions χ_{ι} in H^{2} such that W^{∗} ( \begin{matrix}χ_{ι} \\ η_{ι} \end{matrix} ) ∈ H_{−}^{2}(ℂ^{m_{1}}) ⊕ L^{2}(ℂ^{m_{2}}). We define the

functions ξ_{ι}^{#} ∈ H_{−}^{2}(ℂ^{n_{1}}) ⊕ L^{2}(ℂ^{n_{2}}) by

ξ_{ι}^{#} = A^{t}ξ_{ι} + q_{ι}v,

where q_{ι} is a scalar function in H^{2} satisfying

ℙ_{+}(t_{0}u_{0}q_{ι} + t_{0}u_{0}v^{∗}A^{t}ξ_{ι}) = χ_{ι}

(see the proof of Theorem 6.3).

We can now define M by

M = span{ξ_{ι}^{#} + xv : 1 ≤ ι ≤ N, x ∈ Ker T_{u_{0}}}.

Let us show that dim M = N + k_{0}. Since dim Ker T_{u_{0}} = k_{0}, it is sufficient to prove that if xv + ∑_{ι=1}^{N} c_{ι}ξ_{ι}^{#} = 𝕆, then x = 𝕆 and c_{ι} = 0, 1 ≤ ι ≤ N. This follows immediately from (9.3).

To complete the proof it remains to show that ∥Γ_{Φ}ρ∥_{2} ≥ s∥ρ∥_{2} for ρ = xv + ∑_{ι=1}^{r} c_{j}ξ_{ι}^{#}. Let ξ = ∑_{ι=1}^{r} c_{ι}ξ_{ι}, η = Γ_{Φ}^{(1)}ξ, q = ∑_{ι=1}^{r} c_{ι}q_{ι}, and ξ_{ι}^{#} = ∑_{ι=1}^{r} c_{ι}ξ_{ι}^{#}.

We have

W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) ρ = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) (xv + qv + A^{t}ξ)

= W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) ( \begin{matrix}x + q + v^{∗}A^{t}ξ \\ ξ \end{matrix} )

= W^{∗} ( \begin{matrix}t_{0}u_{0}x + t_{0}u_{0}q + t_{0}u_{0}v^{∗}A^{t}ξ \\ Φ^{(1)}ξ \end{matrix} ) .

It follows (see the proof of Theorem 6.3) that

Γ_{Φ}ρ = W^{∗} ( \begin{matrix}t_{0}u_{0}x + t_{0}u_{0}q + t_{0}u_{0}v^{∗}A^{t}ξ \\ Γ_{Φ^{(1)}}ξ \end{matrix} ) .

Therefore

∥Γ_{Φ}ρ∥_{2}^{2} = |t_{0}|^{2}∥x + q + v^{∗}A^{t}ξ∥_{2}^{2} + ∥η∥_{2}^{2}.

We have

∥ρ∥_{2}^{2} = ∥V^{∗}ρ∥_{2}^{2} = ∥x + q + v^{∗}A^{t}ξ∥_{2}^{2} + ∥ξ∥_{2}^{2}.

Since s ≤ t_{0} and ∥η∥_{2} ≥ s∥ξ∥_{2}, it follows that ∥Γ_{Φ}ρ∥_{2}^{2} ≥ s^{2}∥ρ∥_{2}^{2}. ■

Theorem 10.1 certainly applies to the case of Nehari’s problem. Recall that a matrix function Φ is called very badly approximable (see [PY]) if the zero function is a superoptimal approximant of Φ.

Recall that under the condition ∥H_{Φ}∥_{e} < ∥H_{Φ}∥ the function Φ admits a factorization

Φ = W^{∗} ( \begin{matrix}t_{0}u_{0} & 𝕆 \\ 𝕆 & Φ^{(1)} \end{matrix} ) V^{∗},  (10.2)

where V and W are unitary matrix functions of the form

V = ( v \overline{V}_{c} ) , W = ( w \overline{W}_{c} )^{t} ,

and u_{0} is a unimodular function such that k_{0} =^{def} dim Ker T_{u_{0}} > 0.

The following result is certainly a partial case of Theorem 10.1.

Theorem 10.3.. Let Φ be a very badly approximable matrix function on 𝕋 such that ∥H_{Φ}∥_{e} is less that the smallest nonzero superoptimal singular value of Nehari’s problem. Then

s_{j}(H_{Φ^{(1)}}) ≤ s_{j+k_{0}}(H_{Φ}), j ∈ ℤ_{+},

where Φ^{(1)} and k_{0} are given by the factorization (10.2).

References

[AAK1] V.M. Adamyan, D.Z. Arov, and M.G. Krein, On infinite Hankel matrices and generalized problems of Carathéodory-Fejér and F. Riesz, Funktsional. Anal. i Prilozhen. 2:1 (1968), 1-19.

[AAK2] V.M. Adamyan, D.Z. Arov, and M.G. Krein, On infinite Hankel matrices and generalized problems of Carathéodory-Fejér and I,Schur, Funktsional. Anal. i Prilozhen. 2:2 (1968), 1-17.

[AAK3] V.M. Adamyan, D.Z. Arov, and M.G. Krein, Infinite Hankel block matrices and some related continuation problems, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 6 (1971), 87-112.

[Ar] W.B. Arveson, Interpolation problems in nest algebras, J. Funct. Anal. 20 (1975), 208-233.

[CJ] L. Carleson and S. Jacobs, Best uniform approximation by analytic functions, Ark. Mat. bf 10 (1972), 219-229.

[FT] C. Foias and A. Tannenbaum, On the four block problem, II: the singular system, Int. Equat. Oper. Th. 11 (1988), 726-767.

[Fr] B.A. Francis, A course in H^{∞} control theory, Lecture Notes in Control and Information Sciences 88, Springer-Verlag, Berlin, 1986.

[H] H. Helson, Lectures on invariant subspaces, Acad. Press, NY, 1964.

[Kh] S. Khavinson, On some extremal problems of the theory of analytic functions, Uchen. Zapiski Mosk. Universiteta, Matem. 144:4 (1951), 133-143. English transl.: Amer Math. Soc. Translations (2) 32 (1963), 139-154.

[Ne] Z. Nehari, On bounded bilinear forms, Ann. Math. 65 (1957), 153-162.

[Ni] N.K. Nikol’skii, Treatise on the shift operator, Springer-Verlag, Berlin–Heidelberg–New York– Tokyo, 1986.

[Pa] L.B. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 64 (1970),529-539.

[P] V.V. Peller, Approximation by analytic operator-valued functions, Preprint, 1994.

[PKh] V.V. Peller and S.V. Khrushchev, Hankel operators, best approximation and stationary Gaussian processes, Uspekhi Mat. Nauk 37:1 (1982), 53-124. English Transl.: Russian Math. Surveys 37:1 (1982), 61-144.

[PY1] V.V. Peller and N.J. Young, Superoptimal analytic approximations of matrix functions, J. Funct. Anal. 120 (1994), 300-343.

[PY2] V.V. Peller and N.J. Young, Superoptimal singular values and indices of matrix functions, Int. Eq. Op. Theory

[RSh] W. Rogosinski and H.S. Shapiro, On certain extremum problems for analytic functions, Acta Math. 90 (1953), 287-318.

[R] Yu.A. Rozanov, Stationary stochastic processes, Fizmatgiz, Moscow, 1963. English transl.: Golden-Day, San Francisco, 1967.

[SNF1] B. Sz.-Nagy and C. Foias, Analyse harmonique des opérateurs de l’espace de Hilbert, Akadémiai Kiadó, Budapest, 1967.

[SNF2] B. Sz.-Nagy and C. Foias, On contractions similar to isometries and Toeplitz operators, Ann. Acad. Sci.. Fenn. A I 2 (1976), 553-564.

[T] S.R. Treil, On superoptimal approximation by analytic and meromorphic matrix-valued functions, to appear in J. Funct. Anal.

[TV] S.R. Treil and A.L. Volberg, A fixed point approach to Nehari’s problem and its applications, Oper. Theory: Adv. Appl., Birkhäuser, 71 (1994), 165-186.

[Y] N.J. Young, The Nevanlinna-Pick problem for matrix-valued functions, J. Oper. Theory, 15 (1986), 239-265.

[Va] V. I. Vasyunin, Formula for multiplicity of contractions with finite defect indices, Oper. Theory: Adv. Appl., Birkhäuser, 4 (1989), 281-304.

Vladimir Peller, Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

  E-mail address: peller@math.ksu.edu

Sergei Treil, Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

  E-mail address: treil@math.msu.edu