Matrix valued truncated Toeplitz operators: basic properties

TL;DR

This paper investigates matrix valued truncated Toeplitz operators acting on vector-valued model spaces, generalizing block Toeplitz matrices, and provides characterizations and symbol descriptions for these operators.

Contribution

It offers a characterization of matrix valued truncated Toeplitz operators and identifies symbols that produce the zero operator, extending scalar results to the matrix case.

Findings

Characterization of matrix valued truncated Toeplitz operators

Identification of symbols producing the zero operator

Extension of scalar case results to matrix-valued operators

Abstract

Matrix valued truncated Toeplitz operators act on vector-valued model spaces. They represent a generalization of block Toeplitz matrices. A characterization of these operators analogue to the scalar case is obtained, as well as the determination of the symbols that produce the zero operator.

Equations

$$L^{2}(E)={\Big{\{}F:\mathbb{T}\to E:F(\zeta)=\sum\limits_{-\infty}^{\infty}a_{n}e^{int}:a_{n}\in E,\quad\sum\limits_{-\infty}^{\infty}\|a_{n}\|^{2}<\infty}\Big{\}},$$

$$⟨F,G{⟩}_{L^2(E)}=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}⟨F(e^{it}),G(e^{it}){⟩}_{E}dt.$$

$$L^{\infty }(\mathcal{L}(E))=\{F:\mathbb{T}\to \mathcal{L}(E):\text{F is measurable and bounded}\}$$

$$L^2(\mathcal{L}(E))=[z{H}^2(\mathcal{L}(E)){]}^{\ast }\oplus H^2(\mathcal{L}(E)).$$

$$(S^{\ast }f)(z)=\frac{f(z)-f(0)}{z}.$$

$$k_{\lambda }^{\Theta }(z)=\frac{1}{1-\overline{\lambda }z}(I-\Theta (z)\Theta (\lambda {)}^{\ast }).$$

$$⟨F,k_{\lambda }^{\Theta }x{⟩}_{{\mathcal{K}}_{\Theta }}=⟨F(\lambda ),x{⟩}_E.$$

$$(\tau f)(e^{it})=e^{-it}\Theta (e^{-it}{)}^{\ast }f(e^{-it}),$$

$$({\tau }^{\ast }f)(e^{it})=e^{-it}\Theta (e^{it})f(e^{-it}).$$

$$\widetilde{k_{\lambda}^{\Theta}}x=\frac{1}{z-\lambda}\big{(}\Theta(z)-\Theta(\lambda)\big{)}x.$$

$$(S_{\Theta }f)(z)=P_{\Theta }(zf),f\in {\mathcal{K}}_{\Theta }.$$

$$(S_{\Theta }^{\ast }f)(z)=\frac{f(z)-f(0)}{z};$$

$$\tau S_{\Theta }=S_{\tilde{\Theta }}^{\ast }\tau .$$

$$\begin{split}\mathcal{D}&=\{(I-\Theta(z)\Theta(0)^{*})x:x\in E\}\\ \widetilde{\mathcal{D}}&=\Big{\{}\frac{1}{z}(\Theta(z)-\Theta(0))x:x\in E\Big{\}}.\end{split}$$

$$\begin{array}{rl}(S_{\Theta }f)(z)& =\{\begin{array}{ll}zf(z)& \text{ for }f\perp \tilde{\mathcal{D}},\\ -(I-\Theta (z)\Theta (0{)}^{\ast })\Theta (0)x& \text{ for }f=\frac{1}{z}(\Theta (z)-\Theta (0))x\in \tilde{\mathcal{D}};\end{array}\\ (S_{\Theta }^{\ast }f)(z)& =\{\begin{array}{ll}\frac{f(z)}{z}& \text{ for }f\perp \mathcal{D},\\ -\frac{1}{z}(\Theta (z)-\Theta (0))\Theta (0{)}^{\ast }x.& \text{ for }f=(I-\Theta (z)\Theta (0{)}^{\ast })x\in \mathcal{D},\end{array}\end{array}$$

$$S_{\Theta }(\tilde{\mathcal{D}})\subset \mathcal{D},S_{\Theta }({\tilde{\mathcal{D}}}^{\perp })\subset {\mathcal{D}}^{\perp },S_{\Theta }^{\ast }(\mathcal{D})\subset \tilde{\mathcal{D}},S_{\Theta }^{\ast }({\mathcal{D}}^{\perp })\subset {\tilde{\mathcal{D}}}^{\perp },$$

$$\mathrm{Range}(I-S_{\Theta }{S}_{\Theta }^{\ast })=\mathcal{D},\mathrm{Range}(I-S_{\Theta }^{\ast }{S}_{\Theta })=\tilde{\mathcal{D}},$$

$$(I-S_{\Theta }{S}_{\Theta }^{\ast })f=(I-\Theta (z)\Theta (0{)}^{\ast })f(0).$$

$$P_{\mathcal{D}}$$

$$P_{\tilde{\mathcal{D}}}$$

$$\Omega (I_{{\mathcal{K}}_{\Theta }}-S_{\Theta }{S}_{\Theta }^{\ast })f=(I_{H^2(E)}-S{S}^{\ast })f.$$

$$\begin{array}{rl}{S}_{\Theta }(k_{\lambda }^{\Theta }x)& =\frac{1}{\overline{\lambda }}{k}_{\lambda }^{\Theta }x-\frac{1}{\overline{\lambda }}{k}_0^{\Theta }x\text{for }\lambda \ne 0,\\ S_{\Theta }(\widetilde{k_{\lambda }^{\Theta }}y)& =\lambda \widetilde{k_{\lambda }^{\Theta }}y-k_0^{\Theta }\Theta (\lambda )y.\end{array}$$

$$S_{\Theta }{k}_{\lambda }^{\Theta }x$$

$$\displaystyle=P_{\Theta}\big{(}\frac{z}{1-\overline{\lambda}z}x\big{)}-P_{\Theta}\big{(}\Theta(z)\Theta(\lambda)^{*}\frac{1}{1-\overline{\lambda}z}\big{)}x=P_{\Theta}(\frac{z}{1-\overline{\lambda}z}x)$$

$$\displaystyle=P_{\Theta}\bigg{(}\frac{1}{\overline{\lambda}}\big{(}\frac{1}{1-\overline{\lambda}z}-1\big{)}x\bigg{)}=\frac{1}{\overline{\lambda}}k_{\lambda}^{\Theta}x-\frac{1}{\overline{\lambda}}k_{0}^{\Theta}x.$$

$$S_{\Theta }(\widetilde{k_{\lambda }^{\Theta }}y)$$

$$\displaystyle=P_{\Theta}\big{(}\Theta(z)-\Theta(\lambda)\big{)}y+P_{\Theta}\bigg{(}\frac{\lambda}{z-\lambda}\big{(}\Theta(z)-\Theta(\lambda)\big{)}y\bigg{)}$$

$$=-P_{\Theta }(\Theta (\lambda )y)+\lambda P_{\Theta }\widetilde{k_{\lambda }^{\Theta }}y=\lambda \widetilde{k_{\lambda }^{\Theta }}y-k_0^{\Theta }\Theta (\lambda )y.\text{qed}$$

$$A_{\Phi }^{\ast }=A_{{\Phi }^{\ast }};$$

$$\Theta (z)=\left(\begin{array}{cc}z& 0\\ 0& z^2\end{array}\right),\Phi (z)=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right).$$

$$\Theta H^2(E)=\left\{\left(\begin{array}{c}zf(z)\\ z^{2}g(z)\end{array}\right):f,g\in H^2\right\}.$$

$$\left(\begin{array}{c}z\\ 0\end{array}\right)=\Theta (z)\left(\begin{array}{c}1\\ 0\end{array}\right)\in \Theta H^2(E)\text{but}\left(\begin{array}{c}0\\ z\end{array}\right)=\Phi (z)\left(\begin{array}{c}z\\ 0\end{array}\right)\notin \Theta H^2(E).$$

$$\Phi \Theta =\Theta {\Phi }_1,$$

Full text

1. Introduction

The Toeplitz operators are compressions of multiplication operators on the space $L^{2}(\mathbb{T})$ to the Hardy space $H^{2}$. With respect to the standard exponential basis, their matrices are constant along diagonals; if we truncate such a matrix considering only its upper left finite corner, we obtain classical Toeplitz matrices.

A great deal of attention in the last decade has been attracted by certain generalizations of Toeplitz matrices, namely compressions of multiplication operators to subspaces of the Hardy space which are invariant under the backward shift. These “model spaces” are of the form $H^{2}\ominus uH^{2}$ with $u$ an inner function, and the compressions are called truncated Toeplitz operators. They have been formally introduced in [13]; see [10] for a more recent survey. Although classical Toeplitz matrices have often been a starting point for investigating truncated Toeplitz operators, the latter have a much richer and more interesting theory.

In the theory of contractions on a Hilbert space, these model spaces are the scalar case of a more general construction, which provides functional models for arbitrary completely nonunitary contractions. In particular, it makes sense to consider matrix-valued innner functions $\Theta$ and the associated model space $K_{\Theta}=H^{2}(E)\ominus\Theta H^{2}(E)$, with $E$ a finite dimensional Hilbert space.

We develop below the basics of the corresponding matrix-valued truncated Toeplitz operators, which are compressions to $K_{\Theta}$ of multiplications with matrix-valued functions on $H^{2}(E)$. From an alternate point of view, these operators are generalizations of finite block Toeplitz matrices. With respect to the exposition in [13], one sees that different new questions have to be addressed, mostly related to the noncommutativity of matrices.

The structure of the paper is the following. After a section of general preliminaries about spaces of vector and matrix valued functions, we give a primer of the properties of the vector-valued model space. Matrix-valued truncated Toeplitz operators are introduced in Section 4, where we discuss the main specific difficulties that appear. The next two sections contain the main results of the paper: two intrinsic characterizations of these operators and the identification of symbols that correspond to the null operator. In the final section we determine a class of finite rank matrix-valued truncated Toeplitz operators.

2. Preliminaries

Let $\mathbb{C}$ denote the complex plane, $\mathbb{D}={\{z\in\mathbb{C}:|z|<1}\}$ the unit disc, $\mathbb{T}={\{z\in\mathbb{C}:|z|=1}\}$ the unit circle. Throughout the paper $E$ will denote a fixed Hilbert space, of finite dimension $d$, and ${\mathcal{L}}(E)$ the algebra of bounded linear operators on $E$, which may be identified with $d\times d$ matrices. Part of the development below may be carried through for $E$ an infinite dimensional Hilbert space; we will however restrict ourselves to $\dim E<\infty$, avoiding certain delicate problems of convergence.

The space $L^{2}(E)$ is defined, as usual, by

[TABLE]

endowed with the inner product

[TABLE]

If $\dim E=1$ (i.e $E=\mathbb{C}$) then $L^{2}(E)$ consists of scalar-valued functions and is denoted by $L^{2}$.

The space

[TABLE]

acts on $L^{2}(E)$ by means of multiplication: to $\Phi\in L^{\infty}(\mathcal{L}(E))$ we associate the operator $M_{\Phi}$ defined by $M_{\Phi}(F)=\Phi F$.

By viewing ${\mathcal{L}}(E)$ as a Hilbert space (endowed with the Hilbert–Schmidt norm), one can also consider the space $L^{2}(\mathcal{L}(E))$, which may be identified with matrices with all the entries in $L^{2}(\mathbb{T})$. In particular, $L^{\infty}(\mathcal{L}(E))\subset L^{2}(\mathcal{L}(E))$. Alternately, we may view $L^{2}({\mathcal{L}}(E))$ also as a space of square summable Fourier series with coefficients in $L^{2}(E)$.

The Hardy space $H^{2}(E)$ is the subspace of $L^{2}(E)$ formed by the functions with vanishing negative Fourier coefficients; it can be identified with a space of $E$-valued functions analytic in $\mathbb{D}$, from which the boundary values can be recovered almost everywhere through radial limits. One can also view $H^{2}(E)$ as the direct sum of $d$ standard $H^{2}$ spaces. We have an orthogonal decomposition

[TABLE]

Let $S$ denote the forward shift operator $(Sf)(z)=zf(z)$ on $H^{2}(E)$; it is the restriction of $M_{z}$ to $H^{2}(E)$. Its adjoint (the backward shift) is the operator

[TABLE]

One sees easily that $I-SS^{*}$ is precisely the orthogonal projection onto the space of constant functions.

An inner function will be an element $\Theta\in H^{2}({\mathcal{L}}(E))$ whose boundary values are almost everywhere unitary operators in ${\mathcal{L}}(E)$. The following lemma is a consequence of more general results about factorization of analytic operator valued functions (see [14]).

Lemma 2.1**.**

If $\Theta$ is an inner function, $\Theta_{1}$ is a bounded analytic function, and $\Theta\Theta_{1}$ is constant, then $\Theta$ is also constant.

We will also use the following simple result.

Lemma 2.2**.**

Suppose $\Phi\in H^{2}({\mathcal{L}}(E))$, and $\Phi(z)x\in\Theta H^{2}(E)$ for any $x\in E$. Then there exists $\Phi_{1}\in H^{2}({\mathcal{L}}(E))$ such that $\Phi=\Theta\Phi_{1}$.

Proof.

Let $\{e_{1},\dots,e_{d}\}$ be an orthonormal basis in $E$. Then we may take as $\Phi_{1}$ the matrix having as columns $\Phi(z)e_{1},\dots,\Phi(z)e_{d}$. ∎

The model space associated to $\Theta$, denoted by $\mathcal{K}_{\Theta}$, is defined by $\mathcal{K}_{\Theta}=H^{2}(E)\ominus\Theta H^{2}(E)$; the orthogonal projection onto ${\mathcal{K}}_{\Theta}$ will be denoted by $P_{\Theta}$. The properties of the model space are familiar to many analysts in the scalar case. On the other hand, the vector valued version is less widely known (despite playing an important role in the Sz.-Nagy–Foias theory of contractions [14]); the next section will be a primer of its main properties.

From the point of view of the theory of contractions, the spaces ${\mathcal{K}}_{\Theta}$ represent models for contractions $T$ with $\mathop{\rm rank}(I-T^{*}T)=\mathop{\rm rank}(I-TT^{*})=d$ and $T^{n}\to 0$ strongly. We will not pursue this point of view, which is extensively developed in [14]; rather, we will discuss the model space as an intrinsic functional object.

3. Properties of the model space

For the development in this section we refer to [8], where the context is that of model spaces for completely nonunitary contractions. We will always suppose that the inner function $\Theta$ is pure, which means that $\|\Theta(0)\|<1$.

The model space $\mathcal{K}_{\Theta}$ is a vector valued reproducing kernel Hilbert space; its reproducing kernel function, which takes values in ${\mathcal{L}}(E)$, is

[TABLE]

This means that for any $x\in E$ we have $k_{\lambda}^{\Theta}x\in\mathcal{K}_{\Theta}$, and, if $F\in{\mathcal{K}}_{\Theta}$, then

[TABLE]

If $\Theta(z)\in H^{2}({\mathcal{L}}(E))$ we define the new function $\widetilde{\Theta}(z)=\Theta(\overline{z})^{*}$. Then $\widetilde{\Theta}(z)$ is inner if and only if $\Theta(z)$ is inner. In this case the operator $\tau:L^{2}(E)\to L^{2}(E)$ defined by

[TABLE]

is unitary and $\tau(\mathcal{K}_{\Theta})=\mathcal{K}_{\widetilde{\Theta}}$; thus $\tau P_{\Theta}=P_{\mathcal{K}_{\widetilde{\Theta}}}\tau$. The adjoint of $\tau$ is given by

[TABLE]

We have already met a class of elements in ${\mathcal{K}}_{\Theta}$, namely the functions $k_{\lambda}^{\Theta}x$ for $x\in E$. Another related family is obtained by transporting through $\tau$ the reproducing kernels in $\mathcal{K}_{\widetilde{\Theta}}$. So we define, for $x\in E$, $\widetilde{k_{\lambda}^{\Theta}}x=\tau^{*}(k_{\lambda}^{\widetilde{\Theta}}x)$; computations give

[TABLE]

The model operator $S_{\Theta}\in{\mathcal{L}}(\mathcal{K}_{\Theta})$ is defined by the formula

[TABLE]

The adjoint of $S_{\Theta}$ is given by

[TABLE]

it is the restriction of the left shift in $H^{2}(E)$ to the $S^{*}$-invariant subspace $\mathcal{K}_{\Theta}$. We will also use the formula

[TABLE]

The action of $S_{\Theta}$ is more precisely described if we introduce the following subspaces of $\mathcal{K}_{\Theta}$ (the defect spaces of $S_{\Theta}$ in the terminology of [14]):

[TABLE]

Then the following relations hold:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

From (3.6) it follows that there are operators $J,\tilde{J}\in{\mathcal{L}}({\mathcal{K}}_{\Theta})$, such that

[TABLE]

Since $(I_{H^{2}(E)}-SS^{*})f(z)=f(0)$, $(I_{\mathcal{K}_{\Theta}}-S_{\Theta}S_{\Theta}^{*})f(z)=(I-\Theta(z)\Theta(0)^{*})f(0)$, and $(I-\Theta(z)\Theta(0)^{*})^{-1}\in H^{\infty}({\mathcal{L}}(E))$, we may define the operator $\Omega:{\mathcal{D}}\to E$ by

[TABLE]

The next result is the analog of [13, Lemma 2.2].

Lemma 3.1**.**

For $\lambda\in\mathbb{D}$ and $x,y\in E$ constant functions in $H^{2}(E)$ we have

[TABLE]

Proof.

Since $\Theta(z)\Theta(\lambda)^{*}\frac{1}{1-\overline{\lambda}z}x\in\Theta H^{2}(E)$, we have

[TABLE]

For the second equality, we use $P_{\Theta}(\Theta(z)y)=0$ to obtain

[TABLE]

4. Matrix valued truncated Toeplitz operators

Suppose $\Theta$ is a fixed pure inner function. Since the space $\mathcal{K}_{\Theta}$ is spanned by the functions $k^{\Theta}_{\lambda}x$, $\lambda\in\mathbb{D}$, $x\in E$, which are bounded, it follows that the subspace $\mathcal{K}_{\Theta}^{\infty}=\mathcal{K}_{\Theta}\cap H^{\infty}(E)$ of all bounded functions in $\mathcal{K}_{\Theta}$ is dense in $\mathcal{K}_{\Theta}$.

Suppose now that $\Phi\in L^{2}(\mathcal{L}(E))$. Consider the linear map $f\mapsto P_{\Theta}(\Phi f)$, defined on $\mathcal{K}_{\Theta}^{\infty}$. In case it is bounded, it uniquely determines an operator in ${\mathcal{L}}({\mathcal{K}}_{\Theta})$, denoted by $A_{\Phi}^{\Theta}$, and called a matrix-valued truncated Toeplitz operator (MTTO). The function $\Phi$ is then called a symbol of the operator. We will usually drop the superscript $\Theta$, as we consider a fixed inner function. We denote by ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$ the space of all MTTOs on the model space ${\mathcal{K}}_{\Theta}$.

In the particular case $\Theta(z)=z^{n}I_{E}$, the MTTOs obtained are actually familiar objects, namely block Toeplitz matrices of dimension $n$, in which the entries are matrices of dimension $d$. They have have been extensively studied in linear algebra and related areas (see, for instance, [4]).

It is immediate that

[TABLE]

so ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$ is a selfadjoint linear space.

The operator $S_{\Theta}$ is a simple example of a MTTO; it is obtained by taking $\Phi(z)=zI_{E}$. This example is rather special because the symbol is scalar-valued.

Obviously MTTOs may be viewed as matrix valued analogues of the scalar truncated Toeplitz operators introduced by Sarason in [13]. However, that theory cannot be extended smoothly, with analogous proofs; there are several difficulties that one encounters from the very beginning and that we will point out next.

First, although the space $\Theta H^{2}(E)\subset H^{2}(E)$ is invariant with respect to $S=M_{z}$, it is not invariant for $M_{\Phi}$ for a general analytic $\Phi$, and consequently ${\mathcal{K}}_{\Theta}$ is not invariant with respect to $M_{\Phi}^{*}$; that is, we do not have the relation $A_{\Phi}^{*}=M_{\Phi}^{*}|{\mathcal{K}}_{\Theta}$. This remark is the main source of difficulties in extending the theory from the scalar to the matrix valued case, so it is useful to illustrate it by a simple example.

Example 4.1**.**

Take $d=2$, and

[TABLE]

Then

[TABLE]

So

[TABLE]

This difficulty does not appear in an important particular case. It is easy to prove that if $\Phi\in H^{2}({\mathcal{L}}(E))$ and there exists $\Phi_{1}\in H^{2}(\mathcal{L}(E))$ such that

[TABLE]

then $\Theta H^{2}(E)$ is invariant with respect to $M_{\Phi}$. (In particular, this happens when $\Phi$ commutes with $\Theta$.) Then $M_{\Phi}^{*}{\mathcal{K}}_{\Theta}\subset{\mathcal{K}}_{\Theta}$ and therefore $A_{\Phi}^{*}=M_{\Phi}^{*}|{\mathcal{K}}_{\tilde{\Theta}}\Theta$. It follows that $A_{\Phi}^{*}S_{\Theta}^{*}=S_{\Theta}^{*}A_{\Phi}^{*}$, and therefore $A_{\Phi}S_{\Theta}=S_{\Theta}A_{\Phi}$.

According to the lifting commutant theorem of Sz-Nagy and Foias (see [14, Chapter VI]), the converse is also valid; namely, if $A\in{\mathcal{L}}(E)$ and $AS_{\Theta}=S_{\Theta}A$, then there exists $\Phi$, even in $H^{\infty}({\mathcal{L}}(E))$, such that (4.2) is valid and $A=A_{\Phi}$. A similar result holds by passing to the adjoint and using (4.1). The next theorem yields then the first large class of MTTOs.

Theorem 4.2**.**

The linear space $\{S_{\Theta}\}^{\prime}+\{S_{\Theta}^{*}\}^{\prime}$ is contained in ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$.

Even in the scalar case, the inclusion is in general strict. A simple argument is the fact that, as noted above, the operators in $\{S_{\Theta}\}^{\prime}+\{S_{\Theta}^{*}\}^{\prime}$ always have bounded symbols, which is known not to be in general the case (see [2, 3]).

Another difference from the scalar valued case stems from the nonexistence of a canonical conjugation. Remember that a conjugation on a Hilbert space is a conjugate-linear, isometric and involutive map. If $C$ is a conjugation, then a bounded linear operator $T$ is called $C$-symmetric if $T=CT^{*}C$ (see [9]).

In the scalar case there exists a canonical conjugation with respect to which all truncated Toeplitz operators are symmetric (see [13]). This is no longer true in our case; actually, it follows from results in [5] that the model operator $S_{\Theta}$ is complex symmetric if and only if there exists a conjugation $\Gamma$ on $E$, with the property that for all $z\in\mathbb{D}$ the matrix $\Theta(z)$ is $\Gamma$-symmetric (or, equivalently, $\Theta(e^{it})$ is $\Gamma$-symmetric a.e on $\mathbb{T}$). In that case $C_{\Gamma}$ on $L^{2}(E)$ defined by $C_{\Gamma}f=\Theta\bar{z}\Gamma f$ is a conjugation on $L^{2}(E)$, that leaves ${\mathcal{K}}_{\Theta}$ invariant, and $S_{\Theta}$ is $C_{\Gamma}$-symmetric. However, even if such a conjugation $\Gamma$ exists, not all MTTOs are $C_{\Gamma}$-symmetric. For instance, one can check in Example 4.1 that $\Gamma$ defined on $\mathbb{C}^{2}$ by $\Gamma(a_{1},a_{2})=(\bar{a}_{1},\bar{a}_{2})$ is a conjugation such that $\Theta(z)$ is $\Gamma$-symmetric for all $z\in\mathbb{D}$, but $A_{\Phi}$ is not symmetric with respect to the corresponding $C_{\Gamma}$. The most we can obtain is the following statement, whose proof is straightforward.

Theorem 4.3**.**

Let $\Gamma$ be is a conjugation on $E$, and $C_{\Gamma}f=\Theta\bar{z}\Gamma f$. Suppose that a.e on $\mathbb{T}$ $\Theta(e^{it})$ and $\Phi(e^{it})$ are $\Gamma$-symmetric a.e on $\mathbb{T}$, and $\Phi(e^{it})\Theta(e^{it})=\Theta(e^{it})\Phi(e^{it})$. Then $A_{\Phi}$ is $C_{\Gamma}$-symmetric.

5. Characterization of Matrix Valued Truncated Toeplitz Operators

We obtain characterizations of MTTOs similar to those obtained in the scalar case by Sarason in [13, Theorem 4.1 and 8.1]. We start by recalling that for $\dim E=1$ we have $S_{\Theta}A_{\Phi}=A_{\Phi}S_{\Theta}$, and therefore

[TABLE]

This useful formula is not true in the vector valued context; however, the next lemma provides a useful replacement.

Lemma 5.1**.**

If $\Phi\in H^{2}({\mathcal{L}}(E))$ then

[TABLE]

Proof.

Since $M_{z}\Theta H^{2}(E)\subset\Theta H^{2}(E)$, we have $P_{\Theta}M_{z}P_{\Theta}=P_{\Theta}M_{z}$. Therefore

[TABLE]

Theorem 5.2**.**

The bounded operator $A$ on $\mathcal{K}_{\Theta}$ belongs to $\mathcal{T}(\mathcal{K}_{\Theta})$ if and only if

[TABLE]

for some operators $B,B^{\prime}$ from ${\mathcal{D}}$ to $\mathcal{K}_{\Theta}$.

Proof.

Suppose that A is a bounded operator on $\mathcal{K}_{\Theta}$ that satisfies (5.2). We have then for any positive integer $n$

[TABLE]

and, adding for $n=0,1,2,...,N$,

[TABLE]

Take $f,g\in\mathcal{K}_{\Theta}^{\infty}$; then

[TABLE]

Since $S_{\Theta}^{*N}\to 0$ strongly as $N\to\infty$ we obtain

[TABLE]

or

[TABLE]

Suppose now $f=\sum_{k=0}^{\infty}a_{n}z^{n}$, with $a_{n}\in E$. Then $S_{\Theta}^{*n}f=S^{*n}f=\sum\limits_{k=0}^{\infty}a_{n+k}z^{k}$ and, according to (3.7), $(I-S_{\Theta}S_{\Theta}^{*})S_{\Theta}^{*n}f=(I-\Theta(z)\Theta(0)^{*})a_{n}$. Similarly, if $g=\sum_{k=0}^{\infty}b_{n}z^{n}$, with $b_{n}\in E$, then $(I-S_{\Theta}S_{\Theta}^{*})S_{\Theta}^{*n}g=(I-\Theta(z)\Theta(0)^{*})b_{n}$, whence

[TABLE]

Define then $\Phi,\Phi^{\prime}\in H^{2}({\mathcal{L}}(E))$ by

[TABLE]

Then

[TABLE]

Using the formula $S_{\Theta}^{n}f=P_{\Theta}S^{n}f$ for $f\in\mathcal{K}_{\Theta}$, this becomes

[TABLE]

Therefore $A=A_{\Phi+\Phi^{\prime*}}$, as claimed.

Conversely, suppose that $A=A_{\Phi+\Phi^{\prime*}}$, with $\Phi,\Phi^{\prime}\in H^{2}({\mathcal{L}}(E))$.For $f\in{\mathcal{K}}_{\Theta}^{\infty}$ we have

[TABLE]

According to (5.1), the first term in (5.5) is

[TABLE]

Using the operator $\Omega$ defined by (3.10), we have

[TABLE]

where $B=P_{\Theta}M_{\Phi}\Omega$.

Similarly for $\Phi^{\prime}$, which is also analytic,

[TABLE]

with $B^{\prime}=P_{\Theta}M_{\Phi^{\prime}}\Omega$. Using (5.6) and (5.7) in (5.5), we obtain

[TABLE]

which ends the proof of the theorem. ∎

Remark 5.3**.**

The proof actually shows that for the MTTO $A_{\Phi+\Phi^{\prime*}}$ with $\Phi,\Phi^{\prime}\in H^{2}({\mathcal{L}}(E))$ the operators $B,B^{\prime}$ can be obtained as

[TABLE]

Remark 5.4**.**

An application of the unitary operator $\tau$ defined by (3.1) produces from Theorem 5.1 an alternate necessary and sufficient condition for the bounded operator $A$ to belong to $\mathcal{T}(\mathcal{K}_{\Theta}),$ namely

[TABLE]

for some operators $B,B^{\prime}$ from $\widetilde{\mathcal{D}}$ to $\mathcal{K}_{\Theta}$.

Indeed, one has to consider the operator $\tilde{A}=\tau A\tau^{*}$; then simple computations show that $\tilde{A}\in{\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\tilde{\Theta}})$ if and only if $A\in{\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$, and $\tilde{A}$ satisfies (5.2) on ${\mathcal{K}}_{\tilde{\Theta}}$ if and only $A$ satisfies (5.9) on ${\mathcal{K}}_{\Theta}$.

As in the scalar case, one obtains from Theorem 5.2 a characterization of MTTOs by shift invariance. For a bounded operator on $\mathcal{K}_{\Theta}$ we say that $A$ is shift invariant if

[TABLE]

where $Q_{A}$ is the associated quadratic form on $\mathcal{K}_{\Theta}$, defined by $Q_{A}(f)=\langle Af,f\rangle$ for $f\in\mathcal{K}_{\Theta}$.

Since $Sf\in K_{\Theta}$ if and only if $S_{\Theta}f=zf$, it follows from (3.4) that this is equivalent to $f\in\tilde{\mathcal{D}}^{\perp}$, or to $(I-S_{\Theta}^{*}S_{\Theta})f=0$. If $f\in{\mathcal{K}}^{\infty}_{\Theta}$, then $P_{\tilde{\mathcal{D}}^{\perp}}f\in{\mathcal{K}}^{\infty}_{\Theta}$. Therefore ${\mathcal{K}}^{\infty}_{\Theta}\cap\tilde{\mathcal{D}}^{\perp}$ is dense in $\tilde{\mathcal{D}}^{\perp}$, and the shift invariance condition (5.10) can be checked only for $f\in{\mathcal{K}}^{\infty}_{\Theta}\cap\tilde{\mathcal{D}}^{\perp}$.

Theorem 5.5**.**

A bounded operator A on $\mathcal{K}_{\Theta}$ belongs to $\mathcal{T}(\mathcal{K}_{\Theta})$ if and only if $A$ is shift invariant.

Proof.

Suppose that $A\in\mathcal{T}(\mathcal{K}_{\Theta})$, so $A=A_{\Phi+\Phi^{\prime*}}$ for some $\Phi,\Phi^{\prime}\in H^{2}(\mathcal{L}(E)$. For $f\in{\mathcal{K}}^{\infty}_{\Theta}\cap\tilde{\mathcal{D}}^{\perp}$ we have

[TABLE]

From $f,Sf\in\mathcal{K}_{\Theta}$ it follows that $P_{\Theta}Sf=Sf=zf$. Therefore

[TABLE]

Conversely, suppose that the bounded operator $A$ on $\mathcal{K}_{\Theta}$ is shift invariant. We will prove that it satisfies relation (5.9). Denote $\Delta=A-S_{\Theta}^{*}AS_{\Theta}$. If $f\in\tilde{{\mathcal{D}}}^{\perp}$, then $S_{\Theta}f=Sf$, and

[TABLE]

By the polarization identity we have $\langle\Delta f,g\rangle=0$ for $f,g\in\widetilde{\mathcal{D}}^{\perp}$. Thus the compression of $B$ to $\widetilde{\mathcal{D}}^{\perp}$ is the zero operator, or

[TABLE]

Using (3.9), we obtain

[TABLE]

Therefore $A$ satisfies (5.9), so $A\in{\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$. ∎

Corollary 5.6**.**

The space ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$ is closed in the weak operator topology.

Proof.

Suppose the net $A_{\alpha}\in{\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$ converges weakly to $A$. For $f\in\tilde{{\mathcal{D}}}^{\perp}$ we have, by Theorem 5.5,

[TABLE]

Passing to the limit it follows that

[TABLE]

and the proof is finished by applying again Theorem 5.5. ∎

Remark 5.7**.**

Theorem 5.5 allows us to obtain certain other classes of operators in ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$. Suppose first that $X:\tilde{{\mathcal{D}}}\to{\mathcal{D}}$, and consider $\widehat{X}\in{\mathcal{L}}({\mathcal{K}}_{\Theta})$ defined by $\widehat{X}f=XP_{\tilde{{\mathcal{D}}}}f$. From (3.5) it follows that $f\in\tilde{{\mathcal{D}}}^{\perp}$ then $Sf\in{\mathcal{D}}^{\perp}$ and that $\widehat{X}^{*}f=X^{*}P_{{\mathcal{D}}}f$. Therefore, if $f\in\tilde{{\mathcal{D}}}^{\perp}$, then

[TABLE]

Thus $\widehat{X}$ is shift invariant, hence in ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$ by Theorem 5.5. The operators $\widehat{X}$ have finite rank; we will obtain a more general family of finite rank MTTOs in Section 7.

Further on, the operator

[TABLE]

is also in ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$. The operators $S_{\Theta,X}$ are called modified shifts. For $X$ a contraction, they are precisely the perturbations of $S_{\Theta}$ considered in [8] (and, more generally, in [1]). The case in which $X$ is unitary has been investigated at length in [11]; one obtains then vectorial analogues of the Clark unitary operators introduced in [6].

6. Condition for $A_{\Phi}=0$

We start with the following statement, similar to the scalar case.

Lemma 6.1**.**

If $\Phi\in\Theta H^{2}(\mathcal{L}(E))+[\Theta H^{2}(\mathcal{L}(E))]^{*}$ then $A_{\Phi}=0$.

Proof.

Suppose $\Phi=\Theta\Phi_{1}+\Phi_{2}^{*}\Theta^{*}$, with $\Phi_{1},\Phi_{2}\in\Theta H^{2}(\mathcal{L}(E))$, and $f\in{\mathcal{K}}_{\Theta}^{\infty}$. Then

[TABLE]

Obviously $P_{\Theta}\Theta\Phi_{1}f=0$. On the other side, if we take any $g\in{\mathcal{K}}_{\Theta}^{\infty}$, then

[TABLE]

Therefore $A_{\Phi}=0$. ∎

We are interested to obtain the converse of this result. A first step is the next lemma.

Lemma 6.2**.**

Suppose $A_{\Phi}=0$.

• (i)

If $\Phi\in H^{2}(\mathcal{L}(E))$, then $\Phi=\Theta\Phi_{1}$ for some $\Phi_{1}\in H^{2}(\mathcal{L}(E))$.

• (ii)

If $\Phi^{*}\in H^{2}(\mathcal{L}(E))$, then $\Phi=(\Theta\Phi_{1})^{*}$ for some $\Phi_{1}\in H^{2}(\mathcal{L}(E))$.

Proof.

Clearly it is enough to prove (i), since (ii) follows then by passing to the adjoint.

For any $x\in E$ the function $(I-\Theta(z)\Theta(0)^{*})e_{j}\in{\mathcal{K}}_{\Theta}$, and therefore

[TABLE]

The function $\Phi(z)(I-\Theta(z)\Theta(0)^{*})$ satisfies then the hypothesis of Lemma 2.2. Therefore there exists $G\in H^{2}({\mathcal{L}}(E))$ such that

[TABLE]

or, noting that $(I-\Theta(z)\Theta(0)^{*})^{-1}\in H^{\infty}({\mathcal{L}}(E))$,

[TABLE]

and $G(z)(I-\Theta(z)\Theta(0)^{*})^{-1}\in H^{2}(\mathcal{L}(E))$. ∎

The next result is the desired converse of Lemma 6.1.

Theorem 6.3**.**

If $A_{\Psi}=0$, then there exist $\Psi_{1},\Psi_{2}\in\Theta H^{2}(\mathcal{L}(E))$ such that $\Phi=\Theta\Phi_{1}+\Phi_{2}^{*}\Theta^{*}$.

Proof.

Write $\Psi=\Phi+\Phi^{\prime}{}^{*}$, with $\Phi,\Phi^{\prime}\in H^{2}({\mathcal{L}}(E))$. Applying Theorem 5.2, it follows (see also Remark 5.3) that if $\Omega$ is defined by (3.10) and $B,B^{\prime}$ are given by (5.8), then

[TABLE]

Therefore the range of $B(I-S_{\Theta}S_{\Theta}^{*}))$ is also contained in ${\mathcal{D}}$, so

[TABLE]

Consider the map $\chi:{\mathcal{L}}(E)\to{\mathcal{L}}({\mathcal{D}})$ defined by

[TABLE]

$M_{T}$ being multiplication by the constant operator $T$. We claim that $\chi$ is one-to-one. Indeed, suppose $\chi(T)=0$. Since $\Omega$ is invertible from ${\mathcal{D}}$ to $E$, it follows that $P_{\mathcal{D}}M_{T}=0$, and so

[TABLE]

for all $x,y\in E$. In particular,

[TABLE]

Therefore $Tx\in\Theta H^{2}(E)$. From Lemma 2.2 it follows then that

[TABLE]

for some $G\in H^{\infty}({\mathcal{L}}(E))$. If $T$ is not identically 0, this contradicts Lemma 2.1.

Being a one-to-one map between spaces of the same dimension $d^{2}$, $\chi$ is also onto. Therefore there exists a constant matrix $\Phi_{0}$ such that

[TABLE]

and thus

[TABLE]

Recall now from (6.2) that $P_{\mathcal{D}}M_{\Phi}\Omega P_{\mathcal{D}}=P_{\Theta}M_{\Phi}\Omega P_{\mathcal{D}}$. On the other hand, the values of $M_{\Phi_{0}}\Omega P_{\mathcal{D}}$ are constant functions, and so

[TABLE]

It follows then from (6.3) that

[TABLE]

or

[TABLE]

for any $x\in E$. By Lemma 2.2 there exists $\Phi_{1}\in H^{2}({\mathcal{L}}(E))$ such that $\Phi-\Phi_{0}=\Theta\Phi_{1}$.

In particular, $A_{\Phi-\Phi_{0}}=0$, and therefore, since $A_{\Psi}=A_{\Phi+\Phi^{\prime}{}^{*}}=0$, we also have $A_{(\Phi^{\prime}+\Phi_{0}^{*})^{*}}=0$. Applying Lemma 6.2 (ii), it follows that there exists $\Phi_{2}\in H^{2}({\mathcal{L}}(E))$ such that $(\Phi^{\prime}+\Phi_{0}^{*})^{*}=\Phi_{2}^{*}\Theta^{*}$. Then

[TABLE]

which finishes the proof of the theorem. ∎

As a corollary, we show that every MTTO has a symbol in a certain class. Denote by ${\mathcal{M}}_{\Theta}$ the orthogonal complement of $\Theta H^{2}({\mathcal{L}}(E))$ in $H^{2}({\mathcal{L}}(E))$ endowed with the Hilbert–Schmidt norm. It is easy to see that in a given basis the matrices of functions in ${\mathcal{M}}_{\Theta}$ are characterized by the fact that the columns are functions in ${\mathcal{K}}_{\Theta}$.

Corollary 6.4**.**

For any $A\in{\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$ there exist $\Psi_{1},\Psi_{2}\in{\mathcal{M}}_{\Theta}$ such that $A=A_{\Psi_{1}+\Psi_{2}^{*}}$. If $\Psi_{1}^{\prime},\Psi_{2}^{\prime}\in{\mathcal{M}}_{\Theta}$ also satisfy $A=A_{\Psi^{\prime}_{1}+\Psi^{\prime}_{2}{}^{*}}$, then $\Psi_{1}^{\prime}=\Psi_{1}+k^{\Theta}_{0}X,\Psi_{2}^{\prime}=\Psi_{2}-(k^{\Theta}_{0}X)^{*}$, with $X\in{\mathcal{L}}(E)$.

Proof.

Since $L^{2}({\mathcal{L}}(E))={\mathcal{M}}_{\Theta}+\Theta H^{2}({\mathcal{L}}(E))+{\mathcal{M}}_{\Theta}^{*}+(\Theta H^{2}({\mathcal{L}}(E)))^{*}$, the first assertion follows by decomposing $\Phi$ accordingly and using Theorem 6.3.

For the second part of the corollary, since

[TABLE]

and $\Psi_{1}-\Psi^{\prime}_{1},\Psi_{2}-\Psi^{\prime}_{2}\in{\mathcal{M}}_{\Theta}$, it is enough to show that

[TABLE]

Suppose then that the functions $F,G\in{\mathcal{M}}_{\Theta}$, $F_{1},G_{1}\in H^{2}({\mathcal{L}}(E))$ satisfy

[TABLE]

But we have

[TABLE]

and

[TABLE]

Comparing the last equations, we obtain that $F=(I-\Theta\Theta(0)^{*})X$ for some $X\in{\mathcal{L}}(E)$. Similarly, $G=(I-\Theta\Theta(0)^{*})Y$ for some $Y\in{\mathcal{L}}(E)$. Now, since $F+G^{*}\in\Theta H^{2}+(\Theta H^{2})^{*}$, it follows that the constant matrix $X+Y^{*}$ is in $\Theta H^{2}+(\Theta H^{2})^{*}$.

Suppose $X+Y^{*}\not=0$. Then there are nonzero functions $F_{2},G_{2}\in\Theta H^{2}$ such that $X+Y^{*}=\Theta F_{2}+(\Theta G_{2})^{*}$, or

[TABLE]

Since the left hand side is coanalytic and the right hand side is analytic, both must be constant (and nonzero). Thus $\Theta G_{2}$ is constant, which contradicts Lemma 2.1. Therefore $X=-Y^{*}$, whence (6.4) is satisfied. ∎

It is well known (see, for instance, [12, Chapter 2]), that the space ${\mathcal{K}}_{\Theta}$ is finite dimensional if and only if $\Theta$ is a finite Blaschke–Potapov product. In this case we may use the previous corollary to obtain the dimension of the space ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$.

Corollary 6.5**.**

If $\dim{\mathcal{K}}_{\Theta}=n$, then $\dim{\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})=2n^{d}-d^{2}$.

Proof.

First, it is immediate that $\dim{\mathcal{M}}_{\Theta}=(\dim{\mathcal{K}}_{\Theta})^{d}=n^{d}$. Consider then the linear map $L:{\mathcal{M}}_{\Theta}\times{\mathcal{M}}_{\Theta}\to{\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$ defined by

[TABLE]

According to Corollary 6.4, $L$ is onto, while

[TABLE]

The proof is finished by noting that $\dim({\mathcal{M}}_{\Theta}\times{\mathcal{M}}_{\Theta})=2n^{d}$ and $\dim\ker L=\dim{\mathcal{L}}(E)=d^{2}$. ∎

7. A class of finite rank operators

As noted in Section 3, for any $x\in E$ we have $k^{\Theta}_{\lambda}x\in{\mathcal{K}}_{\Theta}$ and $\widetilde{k^{\Theta}_{\lambda}}\in{\mathcal{K}}_{\Theta}$. Therefore the matrix valued analytic functions $k^{\Theta}_{\lambda}$ and $\widetilde{k^{\Theta}_{\lambda}}$ may be considered as bounded operators from $E$ to ${\mathcal{K}}_{\Theta}$. To avoid any confusion, we will denote these operators by $K_{\lambda},\widetilde{K}_{\lambda}:E\to{\mathcal{K}}_{\Theta}$; therefore $K_{\lambda}^{*},\widetilde{K}_{\lambda}^{*}:{\mathcal{K}}_{\Theta}\to E$. With these notations, the relations in Lemma 3.1 become equalities between operators from $E$ to ${\mathcal{K}}_{\Theta}$:

[TABLE]

We obtain then a class of finite rank operators in ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$.

Theorem 7.1**.**

For any $Y\in{\mathcal{L}}(E)$ and $\lambda\in\mathbb{D}$ the operators $K_{\lambda}Y\widetilde{K}_{\lambda}^{*}$ and $\widetilde{K}_{\lambda}YK_{\lambda}^{*}$ have rank equal to the rank of $Y$ and belong to ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$.

Proof.

Obviously it suffices to consider the first operator. We apply Theorem 5.2. Assuming $\lambda\not=0$ and using (7.1), we obtain

[TABLE]

But the range of $K_{0}$ is contained in ${\mathcal{D}}=\mathop{\rm Range}(I-S_{\Theta}S_{\Theta}^{*})$, and $I-S_{\Theta}S_{\Theta}^{*}$ is invertible on ${\mathcal{D}}$; if we denote that inverse by $Z$, we have $K_{0}=(I-S_{\Theta}S_{\Theta}^{*}))ZK_{0}$, $K_{0}^{*}=K_{0}^{*}Z^{*}(I-S_{\Theta}S_{\Theta}^{*}))$, and

[TABLE]

Equation (5.2) is therefore satisfied by taking $B=\frac{1}{\overline{\lambda}}(K_{\lambda}-K_{0})Y\Theta(\lambda)^{*}K_{0}^{*}Z^{*}$, $B^{\prime}=\widetilde{K}_{\lambda}Y^{*}K_{0}^{*}Z^{*}$. The proof in the case $\lambda\not=0$ is concluded by invoking Theorem 5.2. For $\lambda=0$, we may note that $K_{\lambda}\to K_{0}$ and $\widetilde{K}_{\lambda}\to\widetilde{K}_{0}$ weakly when $\lambda\to 0$, and use Corollary 5.6.

The assertion concerning the rank is left to the reader. ∎

We have thus obtained a class of finite rank MTTOs. For $\lambda=0$, they are precisely the operators $\widehat{X}$ defined in Remark 5.7.

Remark 7.2**.**

In case $Y$ has rank 1, say $Y=x\otimes y$, we have

[TABLE]

and we obtain a family of rank one operators similar to the scalar case. As in the scalar case, we may obtain supplementary operators of rank one in case $k^{\Theta}_{\lambda}x$ and $\widetilde{k^{\Theta}_{\lambda}y}$ have limits in ${\mathcal{K}}_{\Theta}$ when $\lambda$ tends nontangentially to a point $\mu$ on the unit circle. One can show that this is equivalent to the conditions

[TABLE]

Moreover, this procedure allows one to obtain all rank one operators in ${\mathcal{M}}{\mathcal{T}}({\mathcal{K}}_{\Theta})$. The proof is rather tedious and will be presented elsewhere.