m-Isometric block Toeplitz operators

Abstract

In this paper, we study m-isometric block Toeplitz operators with trigonometric symbols. In addition, we give a necessary and sufficient condition for block Toeplitz operators with trigonometric polynomial symbols to be m-contractive.

1 Introduction

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{L(H)}$ be the algebra of all bounded linear operators on $\mathcal{H}$. In the 1990s, Agler and Stankus [1] studied the following operator. For an operator $T\in \mathcal{L}(\mathcal{H})$ and a positive integer m, define

$$ B^{m}(T)=\sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T^{*j}T^{j}. $$

We say that T is m-contractive (respectively, m-expansive and m-isometric) if $B^{m}(T)\ge 0$ (respectively, $B^{m}(T)\leq 0$ and $B^{m}(T)= 0$) for some positive integer m.

The m-isometric operators have been widely investigated in recent years. The theory of m-isometric operators was investigated especially by Agler and Stankus [1,2,3]. Agler and Stankus [1,2,3] developed a rich theory of m-isometric operators and highlighted its connections to Toeplitz operators and function theory. Agler [4] illustrated the connection between m-isometric operators and the classical disconjugacy theory. In recent years, there have been studies on products of m-isometric operators [5], and m-isometric composition operators [6]. In [7], the authors characterized m-isometric Toeplitz operators by properties of the rational symbols and gave some results for m-expansive and m-contractive Toeplitz operators with trigonometric polynomial symbols.

Let $L^{2}\equiv L^{2}(\mathbb{T})$ be the set of square integrable measurable functions on $\mathbb{T}$ and $H^{2}\equiv H^{2}(\mathbb{T})$ be the corresponding Hardy space. Let $L^{\infty } \equiv L^{\infty }( \mathbb{T})$ be the set of bounded measurable functions on $\mathbb{T}$ and let $H^{\infty }\equiv H^{\infty }(\mathbb{T}):=L^{\infty }\cap H^{2}$. We introduce the notion of block Toeplitz operators. Let $M_{n\times r}$ denote the set of all $n\times r$ complex matrices and write $M_{n}=M_{n\times n}$. We observe that $L^{\infty }_{M_{n} }=L^{ \infty }\otimes M_{n}$ and $H^{2}_{\mathbb{C}^{n}}=H^{2}\otimes \mathbb{C} ^{n}$. For the matrix-valued function $\varPhi \in L^{\infty }_{M_{n} }$, the block Toeplitz operator with symbol Φ is the operator $T_{\varPhi }$ on the vector-valued Hardy space $H^{2}_{\mathbb{C}^{n}}$ of the unit disc defined by

$$ T_{\varPhi }f := P_{n}(\varPhi f) \quad \bigl(f\in H^{2}_{\mathbb{C}^{n}}\bigr), $$

where $P_{n}$ denotes the orthogonal projection of $L^{2}_{\mathbb{C} ^{n}}$ ($=L^{2}\otimes \mathbb{C}^{n}$) onto $H^{2}_{\mathbb{C}^{n}}$. If we set $H^{2}_{\mathbb{C}^{n}}=H^{2}(\mathbb{T})\oplus \cdots \oplus H ^{2}(\mathbb{T})$, then we see that if

$$ \varPhi = \begin{bmatrix} \varphi _{11}& \cdots & \varphi _{1n} \\ & \vdots & \\ \varphi _{n1} & \cdots & \varphi _{nn} \end{bmatrix} , \quad \text{then } T_{\varPhi }= \begin{bmatrix} T_{\varphi _{11}}& \cdots & T_{\varphi _{1n}} \\ & \vdots & \\ T_{\varphi _{n1}} & \cdots & T_{\varphi _{nn}} \end{bmatrix} . $$

Gu, Hendricks, and Rutherford [8] studied the hyponormality of block Toeplitz operators and characterized the hyponormality of block Toeplitz operators in terms of their symbols. In particular, they showed that the hyponormality of the block Toeplitz operator $T_{\varPhi }$ will force Φ to be normal, that is, $\varPhi ^{*}\varPhi =\varPhi \varPhi ^{*}$.

This paper is organized as follows. In Sect. 2, we present some preliminary knowledge of block Toeplitz operators and m-isometric operators. In Sect. 3, we give several results for the m-isometric (respectively, m-expansive and m-contractive) block Toeplitz operators with rational symbols. We give a concrete description of m-isometric block Toeplitz operators in terms of the coefficients of the matrix-valued rational symbols.

2 Preliminary

We first review the results of m-isometric Toeplitz operators with rational symbols. The properties of Toeplitz operators enable us to establish several consequences of m-isometric operators. Given a positive integer m, it follows from the definition that an operator $T\in \mathcal{L(H)}$ is an m-isometry if and only if

$$ \sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} \bigl\Vert T^{j}x \bigr\Vert ^{2}=0 \quad \text{for all } x\in \mathcal{H}. $$

(2.1)

Using the identity (2.1) and the block Toeplitz operator with matrix-valued trigonometric polynomial symbol Φ, we consider the following equation:

$$ \sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} \bigl\Vert T_{\varPhi }^{j}K \bigr\Vert ^{2}=0 $$

(2.2)

for all $K\in H^{2}_{\mathbb{C}^{n}}$. A matrix function $\varTheta \in H ^{\infty }_{M_{m\times n}}$ is called inner if $\varTheta ^{*} \varTheta =I_{n}$ a.e. on $\mathbb{T}$. From the well-known results for Toeplitz operators and isometric operators, we can extend the results to block versions. As the idea of the proof is completely the same as in that of [9], we omit the proof of the following lemma.

Lemma 2.1

A necessary and sufficient condition that a Toeplitz operator $T_{\varPhi }$ be an isometric operator is that Φ is inner.

For a matrix-valued function $\varPhi \in H^{2}_{M_{n\times r}}$, write $\widetilde{\varPhi }(z):=\varPhi ^{*}(\overline{z})$. We say that $\Delta \in H^{2}_{M_{n\times m}}$ is a left inner divisor of Φ if Δ is an inner matrix function such that $\varPhi = \Delta A$ for some $A\in H^{2}_{M_{m\times r}} \ (m\leq n)$. We also say that $\varPhi \in H^{2}_{M_{n\times r}}$ and $\varPsi \in H^{2}_{M_{n\times m}}$ are left coprime if the only common left inner divisor of Φ and Ψ is a unitary constant, and that $\varPhi \in H^{2} _{M_{n\times r}}$ and $\varPsi \in H^{2}_{M_{m\times r}}$ are right coprime if Φ̃ and Ψ̃ are left coprime. Two matrix functions Φ and Ψ in $H^{2}_{M_{n}}$ are coprime if they are both left and right coprime.

For $\varPhi \in L^{\infty }_{M_{n}}$, we write

$$ \varPhi _{+}:=P_{n}\varPhi \in H^{2}_{M_{n}} \quad \text{and}\quad \varPhi _{-}:=\bigl(P _{n}^{\perp } \varPhi \bigr)^{*}\in H^{2}_{M_{n}}. $$

Thus we can write $\varPhi =\varPhi _{+}+\varPhi _{-}^{*}$. Recall that a function $\varphi \in L^{\infty }$ is said to be of bounded type (or in the Nevanlinna class) if there are analytic functions $\psi _{1}, \psi _{2}\in H^{\infty }$ such that

$$ \varphi (z)=\frac{\psi _{1}(z)}{\psi _{2}(z)} \quad \text{for almost all } z\in \mathbb{T}. $$

For a matrix-valued function $\varPhi =[\varphi _{ij}]\in L^{\infty }_{M _{n}}$, we say that Φ is of bounded type if each entry $\varphi _{ij}$ is of bounded type and that Φ is rational if each entry $\varphi _{ij}$ is a rational function. Suppose $\varPhi =[\varphi _{ij}]\in L^{\infty }_{M_{n}}$ is such that $\varPhi ^{*}$ is of bounded type. Then we can write $\varphi _{ij}=\theta _{ij} \overline{b}_{ij}$, where $\theta _{ij}$ is inner and $\theta _{ij}$ and $b_{ij}$ are coprime. Thus if θ is the least common multiple of the $\theta _{ij}$, then we write

$$ \varPhi =[\varphi _{ij}]=[\theta _{ij} \overline{b}_{ij}]=[\theta \overline{a}_{ij}]=\varTheta A^{*} $$

(2.3)

where $\varTheta =\theta I_{n}$ and $A\in H^{2}_{M_{n}}$. We note that Eq. (2.3) is minimal, in the sense that if $\omega I_{n}$ (ω is inner) is a common inner divisor of Θ and A, then ω is constant. Let $\varPhi \equiv \varPhi _{+}+\varPhi _{-}^{*}\in L ^{\infty }_{M_{n}}$ be such that Φ and $\varPhi ^{*}$ are of bounded type. Then we can write

$$ \varPhi _{+}=\varTheta _{0}A^{*}\quad \text{and}\quad \varPhi _{-}=\varTheta _{1}B ^{*}, $$

where $\varTheta _{i}=\theta _{i}I_{n}$ with an inner function $\theta _{i}$ for $i=0,1$ and $A,B\in H^{2}_{M_{n}}$. In particular, if $\varPhi \in L^{\infty }_{M_{n}}$ is rational then the $\theta _{i}$ can be chosen as finite Blaschke products. Let Ω be the greatest common left inner divisor of A and $\varTheta _{0}$. Then $A=\varOmega A_{\ell }$ and $\varTheta _{0}=\varOmega \varOmega _{0}$ for some $A_{\ell }\in H^{2}_{M_{n}}$ and some inner matrix $\varOmega _{0}$. Therefore we can write

$$ \varPhi _{+}=A_{\ell }^{*}\varOmega _{0} $$

where $\varOmega _{0}$ and $A_{\ell }$ are left coprime. In this case $\varOmega _{0}A_{\ell }^{*}$ is called a left coprime factorization of $\varPhi _{+}$. Similarly,

$$ \varPhi _{+}=\Delta _{0}A_{r}^{*} $$

where $\Delta _{0}$ and $A_{r}$ are right coprime. In this case $\Delta _{0}A_{r}^{*}$ is called a right coprime factorization of $\varPhi _{+}$.

We write $\mathcal{Z}(\theta )$ is the set of zeros of an inner function θ. The following lemma will be useful in the sequel.

Lemma 2.2

([10])

Let $B\in H^{2}_{M_{n}}$ and $\varTheta :=\theta I_{n}$ with a finite Blaschke product θ. Then the following statements are equivalent:

• (i) $B(\alpha )$ is invertible for each $\alpha \in \mathcal{Z}(\theta )$;

• (ii) B and Θ are right coprime;

• (iii) B and Θ are left coprime.

3 Main results

In this section, we give several results for m-isometric block Toeplitz operators with trigonometric polynomial symbols.

Theorem 3.1

Let $\varPhi :=\varPhi _{+}+\varPhi _{-}^{*}$ is a rational function of the form

$$ \varPhi _{+}=\varTheta _{0}A^{*}\quad \textit{and} \quad \varPhi _{-}=\varTheta _{1}B ^{*}\quad (\textit{coprime factorization}), $$

where $\varTheta _{i}=\theta _{i} I_{n}$ with a finite Blaschke product $\theta _{i}$ for $i=0,1$ and $A,B\in H^{2}_{M_{n}}$ and $\mathcal{Z}( \theta _{0})\cap \mathcal{Z}(\theta _{1})\neq \emptyset$. If $T_{\varPhi }$ is an m-isometry, then Φ is analytic.

Proof

Since $T_{\varPhi }$ is an m-isometry,

$$ \sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T_{\varPhi }^{*j}T_{\varPhi }^{j}K=0 $$

holds for all $K\in H^{2}_{\mathbb{C}^{n}}$. Put $K=\varTheta _{0}^{m}\varTheta _{1}^{m}$. Then

$$\begin{aligned} 0 &=\sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T_{\varPhi }^{*j}T_{\varPhi }^{j}K= \sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} \varPhi ^{*j}\varPhi ^{j}\varTheta _{0}^{m}\varTheta _{1}^{m} \\ &=\sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} \bigl(\varPhi _{+}^{*}+ \varPhi _{-}\bigr)^{j}\bigl(\varPhi _{+}+\varPhi _{-}^{*}\bigr)^{j}\varTheta _{0}^{m} \varTheta _{1}^{m} \\ &=A^{m}B^{m}+H, \end{aligned}$$

for some $H\in \theta H^{\infty }_{M_{n}}$ where $\theta \in \mathcal{Z}(\theta _{0})\cap \mathcal{Z}(\theta _{1})$. Therefore $A^{m}(\alpha )B^{m}(\alpha )=0$ for $\alpha \in \mathcal{Z}(\theta )$. By Lemma 2.2, $A(\alpha )$ and $B(\alpha )$ are invertible, a contradiction. So either $\varPhi _{+}$ or $\varPhi _{-}$ is zero. If $\varPhi _{+}=0$, i.e., $\varPhi =\varPhi _{-}^{*}$, then, for $\varTheta _{1}^{m} \in H^{2}_{\mathbb{C}^{n}}$,

$$\begin{aligned} 0 &=\sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T_{\varPhi }^{*j}T_{\varPhi }^{j} \varTheta _{1}^{m}=\sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T_{\varPhi _{-}}^{j}T_{\varPhi _{-}^{*}}^{j} \varTheta _{1}^{m} \\ &=\sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} \varPhi _{-}^{j}P_{n} \bigl(\varPhi _{-}^{*j}\varTheta _{1}^{m} \bigr)=(-1)^{m}\varTheta _{1}^{m}+KB \end{aligned}$$

for some nonzero $K\in H^{\infty }_{M_{n}}$. Since B and $\varTheta _{1}$ are coprime, we have a contradiction. Therefore $\varPhi _{-}=0$ and hence Φ is analytic. This completes the proof. □

Example 3.2

Suppose that

$$\varPhi (z)= \begin{bmatrix} \frac{2z-1}{1-\frac{1}{2}z}&0 \\ 0&\frac{z-\frac{1}{3}}{1-\frac{1}{3}z} \end{bmatrix} . $$

Then Φ is analytic, but

$$ T_{\varPhi ^{*}}^{2}T_{\varPhi }^{2}-2T_{\varPhi ^{*}}T_{\varPhi } +I= \begin{bmatrix} 9I&0 \\ 0&I \end{bmatrix} \neq 0. $$

Hence $T_{\varPhi }$ is not 2-isometric.

Next, we show that every m-isometric block Toeplitz operators with trigonometric polynomial symbol is an isometry.

Theorem 3.3

Let $\varPhi :=\varPhi _{+}+\varPhi _{-}^{*}$ be normal and rational of the form

$$ \varPhi _{+}=\varTheta _{0}A^{*}\quad \textit{and} \quad \varPhi _{-}=\varTheta _{1}B ^{*}\quad (\textit{coprime factorization}), $$

where $\varTheta _{i}=\theta _{i} I_{n}$ with a finite Blaschke product $\theta _{i}$ for $i=0,1$ and $A,B\in H^{2}_{M_{n}}$ and $\mathcal{Z}( \theta _{0})\cap \mathcal{Z}(\theta _{1})\neq \emptyset$. Then the Toeplitz operator $T_{\varPhi }$ is an m-isometry if and only if Φ is inner.

Proof

Sufficiency is obvious. To prove necessity, if $T_{\varPhi }$ is an m-isometry, then

$$ \sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T_{\varPhi }^{*j}T_{\varPhi }^{j}K=0 $$

holds for all $K\in H^{2}_{\mathbb{C}^{n}}$. Put $K= \varTheta _{0}^{m}\varTheta _{1}^{m}$. Since Φ is normal

$$\begin{aligned} \sum_{j=0}^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T_{\varPhi }^{*j}T_{\varPhi }^{j} \varTheta _{0}^{m}\varTheta _{1}^{m} &=\sum _{j=0} ^{m}(-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} \varPhi ^{*j}{\varPhi }^{j}\varTheta _{0}^{m}\varTheta _{1}^{m} \\ &=\bigl(\varPhi ^{*}\varPhi -I\bigr)^{m}\varTheta _{0}^{m}\varTheta _{1}^{m}. \end{aligned}$$

Thus $(\varPhi ^{*}\varPhi -I)^{m}=0$ and so, $\varPhi ^{*}\varPhi -I$ is nilpotent of order m. Hence the spectrum $\sigma (\varPhi ^{*}\varPhi )=\{1\}$. Thus $\varPhi ^{*}\varPhi =I$. From Lemma 2.1 and Theorem 3.1, Φ is inner. This completes the proof. □

Corollary 3.4

Suppose that Φ, Ψ are normal and rational of the form

$$ \varPhi _{+}=\varTheta _{0}A^{*}\quad \textit{and} \quad \varPhi _{-}=\varTheta _{1}B ^{*}\quad (\textit{coprime factorization}) $$

and

$$ \varPsi _{+}=\varTheta _{2}C^{*}\quad \textit{and} \quad \varPsi _{-}=\varTheta _{3}D ^{*}\quad (\textit{coprime factorization}), $$

where $\varTheta _{i}=\theta _{i} I_{n}$ with a finite Blaschke product $\theta _{i}$ for $i=0,1,2,3$ and $A,B,C,D\in H^{2}_{M_{n}}$ and $\mathcal{Z}(\theta _{0})\cap \mathcal{Z}(\theta _{1})\neq \emptyset$ and $\mathcal{Z}(\theta _{2})\cap \mathcal{Z}(\theta _{3})\neq \emptyset$. Then the following hold.

• (i) If $T_{\varPhi }$ and $T_{\varPsi }$ are m-isometric operators, then $T_{\varPhi }T_{\varPsi }$ and $T_{\varPsi }T_{\varPhi }$ are m-isometric operators.

• (ii) If $T_{\varPhi }$ and $T_{\varPsi }$ are m-isometric operators, then $T_{\varPhi }-T_{\varPsi }$ is an m-isometric operator if and only if $\varPhi ^{*}\varPsi +\varPsi ^{*}\varPhi =I$.

Proof

By Theorem 3.3, $T_{\varPhi }$ and $T_{\varPsi }$ are isometric operators. Therefore $T_{\varPsi }^{*}T_{\varPhi }^{*}T_{\varPhi }T_{\varPsi }=I$, i.e., $T_{\varPhi }T_{\varPsi }$ is isometric and so ΦΨ is normal. Hence $T_{\varPhi }T_{\varPsi }$ is an m-isometric operator. Similarly, $T_{\varPsi }T_{\varPhi }$ is also an m-isometric operator. For (ii), suppose that $T_{\varPhi }-T_{\varPsi }$ is an m-isometric operator. By Theorem 3.3, $T_{\varPhi }-T_{\varPsi }$ is an isometry. Hence

$$\begin{aligned} 0 &=\bigl(T_{\varPhi }^{*}-T_{\varPsi }^{*}\bigr) (T_{\varPhi }-T_{\varPsi })-I=T_{\varPhi }^{*}T _{\varPhi }-T_{\varPhi }^{*}T_{\varPsi }-T_{\varPsi }^{*}T_{\varPhi }+T_{\varPsi }^{*}T _{\varPsi }-I \\ &=I-T_{\varPhi ^{*}\varPsi +\varPsi ^{*}\varPhi }. \end{aligned}$$

Thus, we have $T_{\varPhi ^{*}\varPsi +\varPsi ^{*}\varPhi }=I$ or equivalently, $\varPhi ^{*}\varPsi +\varPsi ^{*}\varPhi =I$. Conversely, if $\varPhi ^{*}\varPsi +\varPsi ^{*}\varPhi =I$, then $T_{\varPhi }-T_{\varPsi }$ is isometry. Hence $T_{\varPhi }-T_{\varPsi }$ is an m-isometric operator. □

Next, we study contractive and expansive Toeplitz operators. We consider m-contractive Toeplitz operators with trigonometric polynomial symbols when $m=1$ and 2.

The next lemma plays a key role in finding necessary and sufficient conditions for m-expansive and m-contractive Toeplitz operators.

Lemma 3.5

([11])

Let A, B, C be complex matrices where A and C are square matrices. Then

$$ \begin{bmatrix} A&B \\ B^{*}&C \end{bmatrix} \geq 0 \quad \Leftrightarrow \quad \textstyle\begin{cases} A\geq 0, \\ B=AW \quad (\textit{for some } W), \\ C\geq W^{*}AW. \end{cases} $$

Moreover, $rank\left[\begin{array}{cc}A& B\\ B^{\ast }& C\end{array}\right]=rankA\iff C=W^{\ast }AW$.

First, we consider the 1-contractive Toeplitz operators.

Theorem 3.6

Suppose that Φ is a matrix-valued function in $L^{\infty }_{ \mathbb{C}^{2}}$ of the form

$$ \varPhi (z)= \begin{bmatrix} a(z)&b(z) \\ c(z)&d(z) \end{bmatrix} . $$

(3.1)

• (i) If $\Vert a \Vert _{\infty }+ \Vert c \Vert _{\infty }>1$, then $T_{\varPhi }$ is contractive if and only if

  $$ T_{b,d}^{*}T_{b,d}-I\geq T_{b,d}^{*}T_{a,c} \bigl(T_{a,c}^{*}T_{a,c}- I\bigr)^{-1}T _{a,c}^{*}T_{b,d}, $$

  where $T_{x,y}=\left[\begin{array}{c}{T}_x\\ T_y\end{array}\right]$.

• (ii) If $\Vert a \Vert _{\infty }+ \Vert c \Vert _{\infty }=1$, then $T_{\varPhi }$ is contractive if and only if $\Vert b \Vert _{\infty }+ \Vert d \Vert _{\infty }\ge 1$.

• (iii) If $\Vert a \Vert _{\infty }+ \Vert c \Vert _{\infty }<1$, then $T_{\varPhi }$ is never contractive.

Proof

From the definition, $T_{\varPhi }$ is contractive if and only if $T_{\varPhi ^{*}}T_{\varPhi } -I\geq 0$ or equivalently,

$$ \begin{bmatrix} T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c}-I &T_{\overline{a}}T_{b}+T _{\overline{c}}T_{d} \\ T_{\overline{b}}T_{a}+T_{\overline{d}}T_{c}&T_{\overline{b}}T_{b}+T _{\overline{d}}T_{d}-I \end{bmatrix} \geq 0. $$

By Lemma 3.5, $T_{\varPhi }$ is contractive if and only if

$$ T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c} \geq I, \qquad T_{\overline{a}}T _{b}+T_{\overline{c}}T_{d}=(T_{\overline{a}}T_{a}+T_{\overline{c}}T _{c}-I)W $$

(3.2)

and

$$ T_{\overline{b}}T_{b}+T_{\overline{d}}T_{d}-I \geq W^{*}(T_{ \overline{a}}T_{b}+T_{\overline{c}}T_{d}) $$

(3.3)

for some W. If $\Vert a \Vert _{\infty }+ \Vert c \Vert _{ \infty }>1$ then $T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c}- I$ is invertible. From Eq. (3.2), we deduce that

$$ W=(T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c}- I)^{-1}(T_{ \overline{a}}T_{b}+T_{\overline{c}}T_{d}). $$

Therefore, by the inequality (3.3),

$$ T_{\overline{b}}T_{b}+T_{\overline{d}}T_{d}-I\geq \bigl(T_{b}^{*}T_{ \overline{a}}^{*}+T_{d}^{*}T_{\overline{c}}^{*} \bigr) (T_{\overline{a}}T _{a}+T_{\overline{c}}T_{c}- I)^{-1}(T_{\overline{a}}T_{b}+T_{ \overline{c}}T_{d}) $$

or equivalently,

$$ T_{b,d}^{*}T_{b,d}-I\geq T_{b,d}^{*}T_{a,c} \bigl(T_{a,c}^{*}T_{a,c}- I\bigr)^{-1}T _{a,c}^{*}T_{b,d}, $$

where $T_{x,y}=\left[\begin{array}{c}{T}_x\\ T_y\end{array}\right]$. If $\Vert a \Vert _{\infty }+ \Vert c \Vert _{\infty }=1$, then $T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c}= I$ and from Eqs. (3.2) and (3.3),

$$ T_{\overline{b}}T_{b}+T_{\overline{d}}T_{d}-I\geq 0. $$

If $\Vert a \Vert _{\infty }+ \Vert c \Vert _{\infty }<1$, then $T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c}< I$. By Eq. (3.2), $T_{\varPhi }$ is never contractive. This completes the proof. □

Using the same arguments of Theorem 3.6 that we can check the following consequence.

Corollary 3.7

Suppose that Φ is a matrix-valued function in $L^{\infty }_{ \mathbb{C}^{2}}$ of the form (3.1).

• (i) If $T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c}< I$, then $T_{\varPhi }$ is expansive if and only if

  $$ T_{\overline{b}}T_{b}+T_{\overline{d}}T_{d}-I\leq \bigl(T_{b}^{*}T_{ \overline{a}}^{*}+T_{d}^{*}T_{\overline{c}}^{*} \bigr) (T_{\overline{a}}T _{a}+T_{\overline{c}}T_{c}- I)^{-1}(T_{\overline{a}}T_{b}+T_{ \overline{c}}T_{d}). $$

• (ii) If $T_{\overline{a}}T_{a}+T_{\overline{c}}T_{c}= I$, then $T_{\varPhi }$ is expansive if and only if $T_{\overline{b}}T _{b}+T_{\overline{d}}T_{d}-I\leq 0$.

• (iii) If $T_{\overline{a}}T_{a}+T_{\overline{c}}T _{c}> I$, then $T_{\varPhi }$ is never expansive.

Corollary 3.8

Suppose that Φ is a matrix-valued function in $L^{\infty }_{ \mathbb{C}^{2}}$ of the form (3.1). Then the following statements hold.

• (i) If $b=0$ and $c=0$, then $T_{\varPhi }$ is contractive if and only if $T_{a}$ and $T_{d}$ are contractive.

• (ii) If $a=0$ and $d=0$, then $T_{\varPhi }$ is contractive if and only if $T_{b}$ and $T_{c}$ are contractive.

Proof

If $b=c=0$, then from Eqs. (3.2) and (3.3), $T_{\varPhi }$ is contractive if and only if $T_{\overline{a}}T_{a} \geq I$ and $T_{\overline{d}}T_{d}\geq I$. □

Corollary 3.9

Suppose that Φ is a matrix-valued function in $L^{\infty }_{ \mathbb{C}^{2}}$ of the form (3.1) where $T_{a}$, $T_{b}$, $T_{c}$, and $T_{d}$ are isometric. Then $T_{\varPhi }$ is contractive if and only if $T_{\overline{a}b+\overline{c}d}$ is expansive.

Proof

From Lemma 2.1 and Theorem 3.6, we have $T_{a \overline{b}+c\overline{d}}T_{\overline{a}b+\overline{c}d}\leq I$ or equivalently, $T_{\overline{a} b+\overline{c}d}$ is expansive. □

Next, we study 2-contractive Toeplitz operators. Suppose that Φ is a matrix-valued function in $L^{\infty }_{\mathbb{C}^{n}}$ of the form

$$ \varPhi (z)= \begin{bmatrix} A(z)&B(z) \\ C(z)&D(z) \end{bmatrix} . $$

Then by a direct calculation we deduce that

$$ B^{2}(T_{\varPhi })=T_{\varPhi ^{*}}^{2}T_{\varPhi }^{2}-2T_{\varPhi ^{*}}T_{\varPhi } +I= \begin{bmatrix} M_{1}&M_{2} \\ M_{2}^{*}&M_{3} \end{bmatrix}, $$

(3.4)

where

$$\begin{aligned} M_{1}={}&T_{A^{*}}^{2}T_{A}^{2}+T_{A^{*}}T_{C^{*}}T_{C}T_{A}+T_{C^{*}}T _{B^{*}}T_{A}^{2}+T_{C^{*}}T_{D^{*}}T_{C}T_{A}+T_{A^{*}}^{2}T_{B}T _{C} \\ & {} +T_{A^{*}}T_{C^{*}}T_{D}T_{C} +T_{C^{*}}T_{B^{*}}T_{B}T_{C}+T_{C^{*}}T _{D^{*}}T_{D}T_{C}-2T_{A^{*}}T_{A}-2T_{C^{*}}T_{C}+I, \\ M_{2}={}&T_{A^{*}}^{2}T_{A}T_{B}+T_{A^{*}}T_{C^{*}}T_{C}T_{B}+T_{C^{*}}T _{B^{*}}T_{A}T_{B}+T_{C^{*}}T_{D^{*}}T_{C}T_{B}+T_{A^{*}}^{2}T_{B}T _{D} \\ & {} +T_{A^{*}}T_{C^{*}}T_{D}^{2}+T_{C^{*}}T_{B^{*}}T_{B}T_{D}+T_{C^{*}}T _{D^{*}}T_{D}^{2}-2T_{A^{*}}T_{B}-2T_{C^{*}}T_{D}, \end{aligned}$$

and

$$\begin{aligned} M_{3} ={}&T_{B^{*}}T_{A^{*}}T_{A}T_{B}+T_{B^{*}}T_{C^{*}}T_{C}T_{B}+T _{D^{*}}T_{B^{*}}T_{A}T_{B}+T_{D^{*}}^{2}T_{C}T_{B}+T_{B^{*}}T_{A^{*}}T _{B}T_{D} \\ &{} +T_{B^{*}}T_{C^{*}}T_{D}^{2}+T_{D^{*}}T_{B^{*}}T_{B}T_{D}+T _{D^{*}}^{2}T_{D}^{2}-2T_{B^{*}}T_{B}-2T_{D^{*}}T_{D}+I. \end{aligned}$$

Since the induced relations are very complicated, in order to consider the 2-contractive or 2-expansive Toeplitz operators, we consider the case with a simple symbol. Recently, m-isometric Toeplitz operators with single-valued symbols were studied in [7].

Lemma 3.10

([7])

Let φ be a rational function. A Toeplitz operator $T_{\varphi }$ is an m-isometry if and only if $T_{\varphi }$ is an isometry.

Theorem 3.11

Suppose that Φ is a matrix-valued rational function in $L^{\infty }_{\mathbb{C}^{n}}$ of the form

$$ \varPhi (z)= \begin{bmatrix} A(z)&B(z) \\ C(z)&D(z) \end{bmatrix}, $$

where $T_{A}$, $T_{B}$, $T_{C}$, and $T_{D}$ are m-isometric operators. Then $T_{\varPhi }$ is 2-contractive if and only if

$$ M_{1}\geq 0,\qquad M_{2}=M_{1}W,\quad \textit{and} \quad M_{3}\geq W^{*}M_{2} $$

(3.5)

for some W where

$$\begin{aligned} & M_{1}=T_{C^{*}B^{*}A^{2}}+T_{C^{*}D^{*}CA}+T_{{A^{*}}^{2}BC}+T_{A^{*}C ^{*}DC}+I, \\ &M_{2}=T_{C^{*}B^{*}AB}+T_{C^{*}D^{*}CB}+T_{{A^{*}}^{2}BD}+T_{A^{*}C ^{*}D^{2}}, \\ &M_{3}=T_{D^{*}B^{*}AB}+T_{{D^{*}}^{2}CB}+T_{B^{*}A^{*}BD}+T_{B^{*}C ^{*}D^{2}}+I. \end{aligned}$$

In particular, suppose that A, B, C, and D are single-valued trigonometric polynomials. If $T_{\varPhi }$ is 2-contractive then

$$ \deg A=\deg D,\qquad 2\deg A=\deg B+\deg C. $$

(3.6)

Proof

From Lemmas 2.1, 3.10, and Eq. (3.4), if follows that $T_{\varPhi }$ is 2-contractive if and only if the relations (3.5) hold for some W. In particular, if A, B, C, and D are single-valued trigonometric polynomials, then from Lemma 3.10, we can set $A(z)= \lambda _{1}z^{k_{a}}$, $B(z)=\lambda _{2}z^{k_{b}}$, $C(z)=\lambda _{3}z ^{k_{c}}$, and $D(z)=\lambda _{4}z^{k_{d}}$ with $\lvert \lambda _{1} \rvert = \lvert \lambda _{2} \rvert = \lvert \lambda _{3} \rvert = \lvert \lambda _{4} \rvert =1$. Hence

$$ M_{1}=T_{\lambda _{1}^{2}\overline{\lambda _{2}\lambda _{3}}z^{2k_{a}-k _{b}-k_{c}}}+T_{\lambda _{1}\overline{\lambda _{4}}z^{k_{a}-k_{d}}}+T _{\overline{\lambda _{1}}^{2}\lambda _{2}\lambda _{3}z^{k_{b}+k_{c}-2k _{a}}} +T_{\overline{\lambda _{1}}\lambda _{4}z^{k_{d}-k_{a}}}+I. $$

From Eq. (3.5),

$$\begin{aligned} M_{1}\geq 0 & \quad \Longleftrightarrow \quad T_{\lambda _{1}^{2}\overline{\lambda _{2}\lambda _{3}}z^{2k_{a}-k_{b}-k_{c}}+\lambda _{1}\overline{\lambda _{4}}z^{k_{a}-k_{d}}+\overline{\lambda _{1}}^{2}\lambda _{2}\lambda _{3}z ^{k_{b}+k_{c}-2k_{a}} +\overline{\lambda _{1}}\lambda _{4}z^{k_{d}-k _{a}}}+I\geq 0 \\ &\quad \Longleftrightarrow\quad T_{2\operatorname{Re}\{\lambda _{1}^{2}\overline{\lambda _{2}\lambda _{3}}z^{2k_{a}-k_{b}-k_{c}}+\lambda _{1}\overline{\lambda _{4}}z^{k_{a}-k_{d}}\}+1}\geq 0, \end{aligned}$$

and hence $k_{a}=k_{d}$ and $2k_{a}=k_{b}+k_{c}$. This completes the proof. □

The next result is a necessary and sufficient condition for $T_{\varPhi }$ to be 2-contractive, under the same hypotheses as Theorem 3.11 but with the additional condition that each entry in the matrix is a single-valued function.

Corollary 3.12

Suppose that Φ is a matrix-valued function in $L^{\infty }_{ \mathbb{C}^{2}}$ of the form

$$ \varPhi (z)= \begin{bmatrix} az^{m}&bz^{l} \\ cz^{2m-l}&dz^{m} \end{bmatrix} $$

where $\lvert a \rvert = \lvert b \rvert = \lvert c \rvert = \lvert d \rvert =1$. Then $T_{\varPhi }$ is 2-contractive if and only if

$$ \textstyle\begin{cases} (2\operatorname{Re}\{\overline{a}^{2}bc+\overline{a}d\}+1)(2\operatorname{Re}\{ \overline{bc}d^{2}+a\overline{d}\}+1)\geq \lvert a\overline{c}+b \overline{d}+\overline{a}^{2}bd+\overline{ac}d^{2} \rvert ^{2} \\ \quad \textit{if } 2\operatorname{Re}\{a^{2}\overline{bc}+a\overline{d}\}+1>0, \\ 2\operatorname{Re}\{\overline{bc}d^{2}+a\overline{d}\}+1\geq 0 \quad \textit{if } 2\operatorname{Re}\{a^{2}\overline{bc}+a\overline{d}\}+1=0. \end{cases} $$

Furthermore, if $2\operatorname{Re}\{a^{2}\overline{bc}+a\overline{d}\}+1<0$, then $T_{\varPhi }$ is not 2-contractive.

Proof

From the proof of Theorem 3.11, we deduced that $T_{\varPhi }$ is 2-contractive if and only if

$$ M_{1}\geq 0,\qquad M_{2}=M_{1}W, \quad \text{and} \quad M_{3}\geq W^{*}M_{2} $$

(3.7)

for some W where

$$\begin{aligned} &M_{1}=T_{a^{2}\overline{bc}}+T_{a\overline{d}}+T_{\overline{a}^{2}bc}+T _{\overline{a}d}+I, \\ &M_{2}=T_{a\overline{c}z^{l-m}}+T_{b\overline{d}z^{l-m}}+T_{ \overline{a}^{2}bdz^{l-m}}+T_{\overline{ac}d^{2}z^{l-m}}, \\ & M_{3}=T_{a\overline{d}}+T_{bc\overline{d}^{2}}+T_{\overline{a}d}+T _{\overline{bc}d^{2}}+I. \end{aligned}$$

From Eq. (3.7), $M_{1}\geq 0$ if and only if $2\operatorname{Re}\{a ^{2}\overline{bc}+a\overline{d}\}+1\geq 0$. There are two cases to consider. If $2\operatorname{Re}\{a^{2}\overline{bc}+a\overline{d}\}+1> 0$, then

$$ W=\bigl(2\operatorname{Re}\bigl\{ a^{2}\overline{bc}+a\overline{d}\bigr\} +1\bigr)^{-1}T_{(a \overline{c}+b\overline{d}+\overline{a}^{2}bd+\overline{ac}d^{2})z ^{l-m}}. $$

Hence $M_{3}\geq W^{*}M_{2}$ if and only if

$$ \bigl(2\operatorname{Re}\bigl\{ a^{2}\overline{bc}+a\overline{d}\bigr\} +1\bigr) \bigl(2\operatorname{Re}\bigl\{ \overline{bc}d^{2}+a \overline{d}\bigr\} +1\bigr) \geq \bigl\lvert a\overline{c}+b \overline{d}+ \overline{a}^{2}bd+\overline{ac}d^{2} \bigr\rvert ^{2}. $$

If $2\operatorname{Re}\{a^{2}\overline{bc}+a\overline{d}\}+1=0$, then $M_{1}=M_{2}=0$ and hence $M_{3}\geq 0$ or equivalently, $2\operatorname{Re} \{\overline{bc}d^{2}+a\overline{d}\}+1\geq 0$. This completes the proof. □

Example 3.13

Suppose that $\Phi (z)=\left[\begin{array}{cc}-z& -z\\ z& z\end{array}\right]$. Then Eq. (3.6) holds, but $M_{1}=-3I<0$. Therefore $T_{\varPhi }$ is not 2-contractive. Hence the converse of Theorem 3.11 does not hold.

4 Conclusion

In [7], m-isometric Toeplitz operators were studied for the case of single trigonometric polynomial symbols. In this paper, we study the properties of Toeplitz operators with matrix-valued trigonometric polynomial symbols and we obtain a necessary and sufficient condition for block Toeplitz operators with trigonometric polynomial symbols to be m-isometric or m-contractive.