A rank formula for the self-commutators of rational Toeplitz tuples

Abstract

In this paper we derive a rank formula for the self-commutators of tuples of Toeplitz operators with matrix-valued rational symbols.

1 Introduction

Let $\mathcal{H}$ and $\mathcal{K}$ be complex Hilbert spaces, let $\mathcal{B(H,K)}$ be the set of bounded linear operators from $\mathcal{H}$ to $\mathcal{K}$, and write $\mathcal{B(H)}:=\mathcal{B(H,H)}$. For $A,B\in\mathcal{B(H)}$, we let $[A,B]:=AB-BA$. An operator $T\in\mathcal{B(H)}$ is said to be normal if $[T^{*},T]=0$, hyponormal if $[T^{*},T]\ge0$. For an operator $T\in\mathcal{B(H)}$, we write kerT and ranT for the kernel and the range of T, respectively. For a subset $\mathcal {M}$ of a Hilbert space $\mathcal {H}$, $\operatorname{cl} \mathcal {M}$ and $\mathcal {M}^{\perp}$ denote the closure and the orthogonal complement of $\mathcal {M}$, respectively. Also, let $\mathbb{T}\equiv\partial\mathbb{D}$ be the unit circle (where $\mathbb {D}$ denotes the open unit disk in the complex plane $\mathbb {C}$). Recall that $L^{\infty}\equiv L^{\infty}(\mathbb {T})$ is the set of bounded measurable functions on $\mathbb {T}$, that the Hilbert space $L^{2}\equiv L^{2}({\mathbb {T}})$ has a canonical orthonormal basis given by the trigonometric functions $e_{n}(z)=z^{n}$, for all $n\in{\mathbb {Z}}$, and that the Hardy space $H^{2}\equiv H^{2}({\mathbb {T}})$ is the closed linear span of $\{e_{n}: n \geq0 \}$. An element $f\in L^{2}$ is said to be analytic if $f\in H^{2}$. Let $H^{\infty}:=L^{\infty}\cap H^{2}$, i.e., $H^{\infty}$ is the set of bounded analytic functions on $\mathbb{D}$.

We review the notion of functions of bounded type and a few essential facts about Hankel and Toeplitz operators and for that we will use [1–4].

For $\varphi\in L^{\infty}$, we write

$$\varphi_{+}\equiv P \varphi\in H^{2} \quad\mbox{and} \quad\varphi _{-} \equiv \overline{P^{\perp}\varphi}\in zH^{2}, $$

where P and $P^{\perp}$ denote the orthogonal projection from $L^{2}$ onto $H^{2}$ and $(H^{2})^{\perp}$, respectively. Thus we may write $\varphi=\overline{\varphi_{-}}+\varphi_{+}$. We recall that a function $\varphi\in L^{\infty}$ is said to be of bounded type (or in the Nevanlinna class $\mathcal {N}$) if there are functions $\psi_{1},\psi_{2}\in H^{\infty}$ such that

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

We recall [5], Lemma 3, that if $\varphi\in L^{\infty}$ then

$$ \varphi\mbox{ is of bounded type} \quad\Longleftrightarrow\quad \operatorname{ker} H_{\varphi}\ne\{0\} . $$

(1.1)

Assume now that both φ and φ̅ are of bounded type. Then from the Beurling’s theorem, $\mbox{ker} H_{\overline{\varphi_{-}}}=\theta_{0} H^{2}$ and $\mbox{ker} H_{\overline{\varphi_{+}}}=\theta_{+} H^{2}$ for some inner functions $\theta_{0}, \theta_{+}$. We thus have $b:={\overline{\varphi_{-}}}\theta_{0} \in H^{2}$, and hence we can write

$$ \varphi_{-}=\theta_{0}\overline{b} \mbox{ and similarly } \varphi_{+} =\theta_{+}\overline{a} \quad\mbox{for some } a \in H^{2}. $$

(1.2)

By Kronecker’s lemma [3], p.183, if $f\in H^{\infty}$ then f̅ is a rational function if and only if $\operatorname{rank} H_{\overline{f}}<\infty$, which implies that

$$ \overline{f}\mbox{ is rational} \quad\Longleftrightarrow\quad f= \theta\overline{b} \mbox{ with a finite Blaschke product } \theta. $$

(1.3)

Let $M_{n\times r}$ denote the set of all $n\times r$ complex matrices and write $M_{n} :=M_{n\times n}$. For $\mathcal {X}$ a Hilbert space, let $L^{2}_{\mathcal {X}} \equiv L^{2}_{\mathcal {X}}(\mathbb {T})$ be the Hilbert space of $\mathcal {X}$-valued norm square-integrable measurable functions on $\mathbb{T}$ and let $L^{\infty}_{\mathcal {X}} \equiv L^{\infty}_{\mathcal {X}}(\mathbb {T})$ be the set of $\mathcal {X}$-valued bounded measurable functions on $\mathbb{T}$. We also let $H^{2}_{\mathcal {X}} \equiv H^{2}_{\mathcal {X}}(\mathbb {T})$ be the corresponding Hardy space and $H^{\infty}_{\mathcal {X}} \equiv H^{\infty}_{\mathcal {X}}(\mathbb {T}) =L^{\infty}_{\mathcal {X}}\cap H^{2}_{\mathcal {X}}$. We observe that $L^{2}_{\mathbb{C}^{n}}= L^{2}\otimes\mathbb{C}^{n}$ and $H^{2}_{\mathbb{C}^{n}}= H^{2}\otimes\mathbb{C}^{n}$.

For a matrix-valued function $\Phi\equiv ( \varphi_{ij} ) \in L^{\infty}_{M_{n}}$, we say that Φ is of bounded type if each entry $\varphi_{ij}$ is of bounded type, and we say that Φ is rational if each entry $\varphi_{ij}$ is a rational function.

Let $\Phi\equiv (\varphi_{ij}) \in L^{\infty}_{M_{n}}$ be such that $\Phi^{*}$ is of bounded type. Then each $\overline{\varphi}_{ij}$ is of bounded type. Thus in view of (1.2), we may write $\varphi_{ij}=\theta_{ij}\overline{b}_{ij}$, where $\theta_{ij}$ is inner and $\theta_{ij}$ and $b_{ij}$ are coprime, in other words, there does not exist a nonconstant inner divisor of $\theta_{ij}$ and $b_{ij}$. Thus if θ is the least common multiple of $\{\theta_{ij}:i,j=1,2, \ldots, n \}$, then we may write

$$ \Phi= ( \varphi_{ij} ) = (\theta_{ij}\overline{b}_{ij} ) = (\theta\overline{a}_{ij} ) \equiv\theta A^{*} \quad \bigl(\mbox{where }A\equiv ( a_{ji} ) \in H^{2}_{M_{n}}\bigr). $$

(1.4)

In particular, $A(\alpha)$ is nonzero whenever $\theta(\alpha)=0$ and $|\alpha|<1$.

For $\Phi\equiv[\varphi_{ij}]\in L^{\infty}_{M_{n}}$, we write

$$\Phi_{+}:= \bigl[P(\varphi_{ij}) \bigr]\in H^{2}_{M_{n}} \quad \mbox{and}\quad \Phi_{-}:= \bigl[P^{\perp}(\varphi_{ij}) \bigr]^{*} \in H^{2}_{M_{n}}. $$

Thus we may write $\Phi=\Phi_{-}^{*}+\Phi_{+}$. However, it will often be convenient to allow the constant term in $\Phi_{-}$. Hence, if there is no confusion we may assume that $\Phi_{-}$ shares the constant term with $\Phi_{+}$: in this case, $\Phi(0) = \Phi_{+}(0) + \Phi_{-}(0)^{*}$. If $\Phi=\Phi_{-}^{*}+\Phi_{+}\in L^{\infty}_{M_{n}}$ is such that Φ and $\Phi^{*}$ are of bounded type, then in view of (1.4), we may write

$$ \Phi_{+}= \theta_{1} A^{*} \quad\mbox{and}\quad \Phi_{-}= \theta_{2} B^{*}, $$

(1.5)

where $\theta_{1}$ and $\theta_{2}$ are inner functions and $A,B\in H^{2}_{M_{n}}$. In particular, if $\Phi\in L^{\infty}_{M_{n}}$ is rational then the $\theta_{i}$ can be chosen as finite Blaschke products, as we observed in (1.3). For simplicity, we write $H_{0}^{2}$ for $zH^{2}_{M_{n}}$.

We now introduce the notion of Hankel operators and Toeplitz operators with matrix-valued symbols. If Φ is a matrix-valued function in $L^{\infty}_{M_{n}}$, then $T_{\Phi}: H^{2}_{\mathbb{C}^{n}}\to H^{2}_{\mathbb{C}^{n}}$ denotes Toeplitz operator with symbol Φ defined by

$$T_{\Phi}f:=P_{n}(\Phi f) \quad\mbox{for } f\in H^{2}_{\mathbb{C}^{n}}, $$

where $P_{n}$ is the orthogonal projection of $L^{2}_{\mathbb{C}^{n}}$ onto $H^{2}_{\mathbb{C}^{n}}$. A Hankel operator with symbol $\Phi\in L^{\infty}_{M_{n}}$ is an operator $H_{\Phi}: H^{2}_{\mathbb{C}^{n}}\to H^{2}_{\mathbb{C}^{n}}$ defined by

$$H_{\Phi}f := J_{n} P_{n}^{\perp}(\Phi f) \quad\mbox{for } f\in H^{2}_{\mathbb{C}^{n}}, $$

where $P_{n}^{\perp}$ is the orthogonal projection of $L^{2}_{\mathbb{C}^{n}}$ onto $(H^{2}_{\mathbb{C}^{n}})^{\perp}$ and $J_{n}$ denotes the unitary operator from $L^{2}_{\mathbb{C}^{n}}$ onto $L^{2}_{\mathbb{C}^{n}}$ given by $J_{n}(f)(z):= \overline{z} f(\overline{z})$ for $f \in L^{2}_{\mathbb{C}^{n}}$. For $\Phi\in L^{\infty}_{M_{n\times m}}$, write

$$\widetilde{\Phi}(z):=\Phi^{*}(\overline{z}). $$

A matrix-valued function $\Theta \in H^{\infty}_{M_{n\times m}}$ is called inner if $\Theta^{*}\Theta=I_{m}$ almost everywhere on $\mathbb{T}$, where $I_{m}$ denotes the $m\times m$ identity matrix. If there is no confusion we write simply I for $I_{m}$. The following basic relations can easily be derived:

$$\begin{aligned} &T_{\Phi}^{*}=T_{\Phi^{*}}, \qquad H_{\Phi}^{*}= H_{\widetilde{\Phi}} \quad\bigl(\Phi\in L^{\infty}_{M_{n}} \bigr); \end{aligned}$$

(1.6)

$$\begin{aligned} &T_{\Phi\Psi}-T_{\Phi}T_{\Psi}= H_{\Phi^{*}}^{*}H_{\Psi}\quad \bigl(\Phi,\Psi\in L^{\infty}_{M_{n}}\bigr); \end{aligned}$$

(1.7)

$$\begin{aligned} &H_{\Phi}T_{\Psi}= H_{\Phi\Psi}, \qquad H_{\Psi\Phi}=T_{\widetilde{\Psi}}^{*}H_{\Phi}\quad \bigl(\Phi\in L^{\infty}_{M_{n}}, \Psi\in H^{\infty}_{M_{n}} \bigr). \end{aligned}$$

(1.8)

In 2006, Gu et al. [6] have considered the hyponormality of Toeplitz operators with matrix-valued symbols and characterized it in terms of their symbols.

Lemma 1.1

(Hyponormality of block Toeplitz operators [6])

For each $\Phi\in L^{\infty}_{M_{n}}$, let

$$\mathcal{E}(\Phi):= \bigl\{ K\in H^{\infty}_{M_{n}}: \|K \|_{\infty}\le1 \textit{ and } \Phi-K \Phi^{*}\in H^{\infty}_{M_{n}} \bigr\} . $$

Then $T_{\Phi}$ is hyponormal if and only if Φ is normal and $\mathcal{E}(\Phi)$ is nonempty.

For a matrix-valued function $\Phi\in H^{2}_{M_{n\times r}}$, 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 $\Phi=\Delta A$ for some $A \in H^{2}_{M_{m\times r}}$. We also say that two matrix functions $\Phi\in H^{2}_{M_{n\times r}}$ and $\Psi\in H^{2}_{M_{n\times m}}$ are left coprime if the only common left inner divisor of both Φ and Ψ is a unitary constant, and that $\Phi\in H^{2}_{M_{n\times r}}$ and $\Psi\in H^{2}_{M_{m\times r}}$ are right coprime if Φ̃ and Ψ̃ are left coprime. Two matrix functions Φ and Ψ in $H^{2}_{M_{n}}$ are said to be coprime if they are both left and right coprime. We note that if $\Phi\in H^{2}_{M_{n}}$ is such that $\operatorname{det} \Phi\ne0$, then any left inner divisor Δ of Φ is square, i.e., $\Delta\in H^{2}_{M_{n}}$ (cf. [7]). If $\Phi\in H^{2}_{M_{n}}$ is such that $\operatorname{det} \Phi\ne0$, then we say that $\Delta\in H^{2}_{M_{n}}$ is a right inner divisor of Φ if Δ̃ is a left inner divisor of Φ̃.

Let $\{\Theta_{i}\in H^{\infty}_{M_{n}}: i\in J\}$ be a family of inner matrix functions. The greatest common left inner divisor $\Theta_{d}$ and the least common left inner multiple $\Theta_{m}$ of the family $\{\Theta_{i}\in H^{\infty}_{M_{n}}: i\in J\}$ are the inner functions defined by

$$\Theta_{d} H^{2}_{\mathbb {C}^{p}}=\bigvee _{i \in J}\Theta_{i}H^{2}_{\mathbb {C}^{n}} \quad \mbox{and}\quad \Theta_{m} H^{2}_{\mathbb {C}^{q}}=\bigcap _{i\in J}\Theta_{i}H^{2}_{\mathbb {C}^{n}}. $$

Similarly, the greatest common right inner divisor $\Theta_{d}^{\prime}$ and the least common right inner multiple $\Theta_{m}^{\prime}$ of the family $\{\Theta_{i}\in H^{\infty}_{M_{n}}: i\in J\}$ are the inner functions defined by

$$\widetilde{\Theta}_{d}^{\prime} H^{2}_{\mathbb {C}^{r}}= \bigvee_{i \in J}\widetilde{\Theta}_{i} H^{2}_{\mathbb {C}^{n}} \quad\mbox{and}\quad \widetilde{ \Theta}_{m}^{\prime} H^{2}_{\mathbb {C}^{s}}=\bigcap _{i \in J} \widetilde{\Theta}_{i}H^{2}_{\mathbb {C}^{n}}. $$

The Beurling-Lax-Halmos theorem guarantees that $\Theta_{d}$ and $\Theta_{m}$ exist and are unique up to a unitary constant right factor, and $\Theta_{d}^{\prime}$ and $\Theta_{m}^{\prime}$ are unique up to a unitary constant left factor. We write

$$\begin{aligned} &\Theta_{d} =\mbox{left-g.c.d.} \{\Theta_{i}: i\in J\}, \qquad \Theta_{m}=\mbox{left-l.c.m.} \{\Theta_{i}: i\in J\}, \\ &\Theta_{d}^{\prime}=\mbox{right-g.c.d.} \{ \Theta_{i}: i\in J\},\qquad \Theta_{m}^{\prime}= \mbox{right-l.c.m.} \{\Theta_{i}: i\in J\}. \end{aligned}$$

If $n=1$, then $\mbox{left-g.c.d.} \{\cdot\}=\mbox{right-g.c.d.} \{\cdot\}$ (simply denoted $\mbox{g.c.d.} \{\cdot\}$) and $\mbox{left-l.c.m.} \{\cdot\}=\mbox{right-l.c.m.} \{\cdot\}$ (simply denoted $\mbox{l.c.m.} \{\cdot\}$). In general, it is not true that $\mbox{left-g.c.d.} \{\cdot\}=\mbox{right-g.c.d.} \{\cdot\}$ and $\mbox{left-l.c.m.} \{\cdot\}=\mbox{right-l.c.m.} \{\cdot\}$.

If θ is an inner function we write $I_{\theta}$ for $\theta I_{n}$ and $\mathcal{Z}(\theta)$ for the set of all zeros of θ.

Lemma 1.2

Let $\Theta_{i}:=I_{\theta_{i}}$ for an inner function $\theta_{i}$ ($i \in J$).

• (a) $\textit{left-g.c.d.} \{\Theta_{i}: i\in J\}= \textit{right-g.c.d.} \{\Theta_{i}: i\in J\} =I_{\theta_{d}}$, where $\theta_{d}=\textit{g.c.d.} \{\theta_{i} : i \in J \}$.

• (b) $\textit{left-l.c.m.} \{\Theta_{i}: i\in J\}= \textit{right-l.c.m.} \{\Theta_{i}: i\in J\} =I_{\theta_{m}}$, where $\theta_{m}=\textit{l.c.m.} \{\theta_{i} : i \in J \}$.

Proof

See [7], Lemma 2.1. □

In view of Lemma 1.2, if $\Theta_{i}=I_{\theta_{i}}$ for an inner function $\theta_{i}$ ($i\in J$), we can define the greatest common inner divisor $\Theta_{d}$ and the least common inner multiple $\Theta_{m}$ of the $\Theta_{i}$ by

$$\Theta_{d}\equiv\mbox{g.c.d.} \{\Theta_{i}:i\in J \}:=I_{\theta_{d}}, \quad\mbox{where } \theta_{d}=\mbox{g.c.d.} \{ \theta_{i} : i \in J \} $$

and

$$\Theta_{m}\equiv\mbox{l.c.m.} \{\Theta_{i}:i\in J \}:=I_{\theta_{m}}, \quad\mbox{where } \theta_{m}=\mbox{l.c.m.} \{ \theta_{i} : i \in J \}. $$

Both $\Theta_{d}$ and $\Theta_{m}$ are diagonal-constant inner functions, i.e., diagonal inner functions, and constant along the diagonal.

By contrast with scalar-valued functions, in (1.4), $I_{\theta}$ and A need not be (right) coprime. If $\Omega=\mbox{left-g.c.d.} \{I_{\theta}, A\}$ in the representation (1.4), that is,

$$\Phi=\theta A^{*} , $$

then $I_{\theta}=\Omega\Omega_{\ell}$ and $A=\Omega A_{\ell}$ for some inner matrix $\Omega_{\ell}$ (where $\Omega_{\ell}\in H^{2}_{M_{n}}$ because $\operatorname{det} (I_{\theta})\neq0$) and some $A_{l} \in H^{2}_{M_{n}}$. Therefore if $\Phi^{*}\in L^{\infty}_{M_{n}}$ is of bounded type then we can write

$$ \Phi={A_{\ell}}^{*}\Omega_{\ell}, \quad\mbox{where }A_{\ell}\mbox{ and } \Omega_{\ell}\mbox{ are left coprime}. $$

(1.9)

In this case, $A_{\ell}^{*}\Omega_{\ell}$ is called the left coprime factorization of Φ and write, briefly,

$$ \Phi=A_{\ell}^{*}\Omega_{\ell} \quad (\mbox{left coprime}). $$

(1.10)

Similarly, we can write

$$ \Phi=\Omega_{r} A_{r}^{*}, \quad\mbox{where }A_{r}\mbox{ and }\Omega_{r}\mbox{ are right coprime}. $$

(1.11)

In this case, $\Omega_{r} A_{r}^{*}$ is called the right coprime factorization of Φ and we write, succinctly,

$$ \Phi=\Omega_{r} A_{r}^{*} \quad (\mbox{right coprime}). $$

(1.12)

In this case, we define the degree of Φ by

$$\operatorname{deg} (\Phi):=\dim\mathcal{H}(\Omega_{r}), $$

where $\mathcal{H}(\Theta):=H^{2}_{\mathbb {C}^{n}}\ominus \Theta H^{2}_{\mathbb {C}^{n}}$ for an inner function Θ. It was known (cf. [8], Lemma 3.3) that if θ is a finite Blaschke product then $I_{\theta}$ and $A\in H^{2}_{M_{n}}$ are left coprime if and only if they are right coprime. In this viewpoint, in (1.10) and (1.12), $\Omega_{\ell}$ or $\Omega_{r}$ is $I_{\theta}$ (θ a finite Blaschke product) then we shall write

$$\Phi=\theta A^{*}\quad (\mbox{coprime}). $$

On the other hand, we recall that an operator $T\in\mathcal{B(H)}$ is said to be subnormal if T has a normal extension, i.e., $T=N\vert_{\mathcal{H}}$, where N is a normal operator on some Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ such that $\mathcal {H}$ is invariant for N. The Bram-Halmos criterion for subnormality [9, 10] states that an operator $T\in \mathcal{B(H)}$ is subnormal if and only if $\sum_{i,j}(T^{i}x_{j}, T^{j} x_{i})\ge0$ for all finite collections $x_{0},x_{1},\ldots,x_{k}\in\mathcal{H}$. It is easy to see that this is equivalent to the following positivity test:

$$ \begin{pmatrix} [T^{*},T]& [T^{*2},T]& \ldots& [T^{*k},T]\\ [T^{*}, T^{2}]& [T^{*2},T^{2}] & \ldots& [T^{*k},T^{2}]\\ \vdots& \vdots& \ddots& \vdots\\ [T^{*}, T^{k}] & [T^{*2}, T^{k}] & \ldots& [T^{*k},T^{k}] \end{pmatrix} \ge0 \quad(\mbox{all }k\ge1) . $$

(1.13)

Condition (1.13) provides a measure of the gap between hyponormality and subnormality. In fact the positivity condition (1.13) for $k=1$ is equivalent to the hyponormality of T, while subnormality requires the validity of (1.13) for all k. For $k\ge1$, an operator T is said to be k-hyponormal if T satisfies the positivity condition (1.13) for a fixed k. Thus the Bram-Halmos criterion can be stated thus: T is subnormal if and only if T is k-hyponormal for all $k\ge1$. The notion of k-hyponormality has been considered by many authors aiming at understanding the bridge between hyponormality and subnormality. In view of (1.13), between hyponormality and subnormality there exists a whole slew of increasingly stricter conditions, each expressible in terms of the joint hyponormality of the tuples $(I,T,T ^{2},\ldots,T^{k})$. Given an n-tuple $\mathbf {T}=(T_{1},\ldots, T_{n})$ of operators on $\mathcal{H}$, we let $[\mathbf {T}^{*},\mathbf {T}] \in\mathcal{B(H\oplus\cdots\oplus H)}$ denote the self-commutator of T, defined by

$$\bigl[\mathbf {T}^{*}, \mathbf {T}\bigr]:= \begin{pmatrix} [T_{1}^{*}, T_{1}]& [T_{2} ^{*}, T_{1}]& \ldots& [T_{n}^{*},T_{1}]\\ [T_{1}^{*}, T_{2}]&[T_{2} ^{*}, T_{2}]& \ldots& [T_{n}^{*},T_{2}]\\ \vdots& \vdots& \ddots& \vdots\\ [T_{1}^{*},T_{n}] & [T_{2}^{*},T_{n}] & \ldots& [T_{n}^{*},T_{n}] \end{pmatrix}. $$

By analogy with the case $n=1$, we shall say [11, 12] that T is jointly hyponormal (or simply, hyponormal) if $[\mathbf {T}^{*},\mathbf {T}]\ge0$, i.e., $[\mathbf {T}^{*},\mathbf {T}]$ is a positive-semidefinite operator on $\mathcal{H} \oplus\cdots\oplus\mathcal{H}$.

Tuples $\mathbf{T}\equiv(T_{\Phi_{1}}, \ldots, T_{\Phi_{m}})$ of block Toeplitz operators $T_{\Phi_{i}}$ ($i=1,\ldots,m$) will be called a (block) Toeplitz tuples. Moreover, if each Toeplitz operator $T_{\Phi_{i}}$ has a symbol $\Phi _{i}$ which is a matrix-valued rational function, then the tuple $\mathbf{T}\equiv(T_{\Phi_{1}}, \ldots, T_{\Phi_{m}})$ is called a rational Toeplitz tuple. In this paper we will derive a rank formula for the self-commutator of a rational Topelitz tuple.

2 The results and discussion

For an operator $S\in\mathcal{B(H)}$, $S^{\sharp} \in\mathcal{B(H)}$ is called the Moore-Penrose inverse of S if

$$SS^{\sharp}S=S,\qquad S^{\sharp}SS^{\sharp}=S^{\sharp}, \qquad \bigl(S^{\sharp}S\bigr)^{*}= S^{\sharp}S, \quad\mbox{and} \quad \bigl(SS^{\sharp}\bigr)^{*}=SS^{\sharp}. $$

It is well known [13], Theorem 8.7.2, that if an operator S on a Hilbert space has a closed range then S has a Moore-Penrose inverse. Moreover, the Moore-Penrose inverse is unique whenever it exists. On the other hand, it is well known that if

$$S:= \begin{bmatrix} A&B\\ B^{*}&C \end{bmatrix} \quad\mbox{on } \mathcal{H}_{1}\oplus\mathcal{H}_{2} $$

(where the $\mathcal {H}_{j}$ are Hilbert spaces, $A\in \mathcal{B}(\mathcal {H}_{1})$, $C\in\mathcal{B}(\mathcal {H}_{2})$, and $B\in\mathcal{B}(\mathcal {H}_{2}, \mathcal {H}_{1})$), then

$$ S\ge0 \quad\Longleftrightarrow\quad A\ge0, C\ge0, \mbox{ and } B=A^{\frac{1}{2}}DC^{\frac{1}{2}} \quad\mbox{for some contraction } D; $$

(2.1)

moreover, in [14], Lemma 1.2, and [15], Lemma 2.1, it was shown that if $A\ge0$, $C\ge0$, and ranA is closed then

$$ S\ge0 \quad\Longleftrightarrow\quad B^{*}A^{\sharp}B\le C \mbox{ and } \operatorname{ran} B\subseteq\operatorname{ran} A, $$

(2.2)

or equivalently [12], Lemma 1.4,

$$ \bigl|\langle Bg, f\rangle \bigr|^{2}\le\langle Af,f\rangle \langle Cg, g\rangle \quad\mbox{for all } f\in\mathcal{H}_{1}, g\in \mathcal{H}_{2} $$

(2.3)

and furthermore, if both A and C are of finite rank then

$$ \operatorname{rank} S=\operatorname{rank} A+\operatorname{rank} \bigl(C-B^{*}A^{\sharp}B\bigr). $$

(2.4)

In fact, if $A\ge0$ and ranA is closed then we can write

$$A= \begin{bmatrix} A_{0}&0\\ 0&0 \end{bmatrix} : \begin{bmatrix} \operatorname{ran} A\\ \operatorname{ker} A \end{bmatrix} \to \begin{bmatrix} \operatorname{ran} A\\ \operatorname{ker} A \end{bmatrix} , $$

so that the Moore-Penrose inverse of A is given by

$$ A^{\sharp}= \begin{bmatrix} (A_{0})^{-1}&0\\ 0&0 \end{bmatrix} . $$

(2.5)

Proposition 2.1

If $A\in\mathcal{B(H)}$ has a closed range then $A(A^{*}A)^{\sharp}A^{*}$ is the orthogonal projection onto ranA.

Proof

Suppose $A\in\mathcal{B(H)}$ has a closed range. Then (2.5) can be written as

$$ (P_{\operatorname{ran} A} A P_{\operatorname{ran} A} )^{-1} =P_{\operatorname{ran} A} A^{\sharp}P_{\operatorname{ran} A}. $$

(2.6)

Since by assumption, $A^{*}A$ has also a closed range, there exists the Moore-Penrose inverse $(A^{*}A)^{\sharp}$. Observe

$$\bigl(A\bigl(A^{*}A\bigr)^{\sharp}A^{*}\bigr) \bigl(A\bigl(A^{*}A \bigr)^{\sharp}A^{*}\bigr)=A\bigl(A^{*}A\bigr)^{\sharp}A^{*} $$

and

$$\bigl(A\bigl(A^{*}A\bigr)^{\sharp}A^{*}\bigr)^{*}=A\bigl(A^{*}A \bigr)^{\sharp}A^{*} , $$

which implies that $A(A^{*}A)^{\sharp}A^{*}$ is an orthogonal projection. Put

$$K:=\operatorname{ran} A^{*}A=\operatorname{ran} A^{*}=(\operatorname{ker} A)^{\perp}. $$

We then have

$$\begin{aligned} A\bigl(A^{*}A\bigr)^{\sharp}A^{*} &=AP_{K}\bigl(A^{*}A \bigr)^{\sharp}P_{K} A^{*} \\ &=A\bigl(P_{K}\bigl(A^{*}A\bigr)P_{K}\bigr)^{-1}A^{*} \quad\bigl(\mbox{by }(2.5)\bigr) , \end{aligned}$$

which implies that $\operatorname{ran} (A(A^{*}A)^{\sharp}A^{*} ) =\operatorname{ran} A$. □

In the sequel we often encounter the following matrix:

$$S:= \begin{bmatrix} A^{*}A&A^{*}B\\ B^{*}A&[B^{*},B] \end{bmatrix} , $$

where A has a closed range. If $S\ge0$ and if A and $[B^{*}, B]$ are of finite rank then by (2.4), we have

$$ \operatorname{rank} S=\operatorname{rank} \bigl(A^{*}A\bigr)+ \operatorname{rank} \bigl( \bigl[B^{*},B\bigr]-B^{*}A\bigl(A^{*}A\bigr)^{\sharp}A^{*} B \bigr). $$

(2.7)

Thus, if we write $P_{K}$ for the orthogonal projection onto $K:= \operatorname{ran} A$, then by Proposition 2.1 we have

$$ \begin{aligned}[b] \operatorname{rank} S &=\operatorname{rank} \bigl(A^{*}\bigr)+\operatorname{rank} \bigl(\bigl[B^{*},B\bigr]-B^{*}P_{K} B \bigr) \\ &=\operatorname{rank} \bigl(A^{*}\bigr)+\operatorname{rank} \bigl(B^{*}P_{K^{\perp}}B-BB^{*} \bigr) . \end{aligned} $$

(2.8)

If $\Phi, \Psi\in L^{\infty}_{M_{n}}$, then by (1.7),

$$[T_{\Phi}, T_{\Psi}]= H_{\Psi^{*}}^{*} H_{\Phi} - H_{\Phi^{*}}^{*}H_{\Psi}+ T_{\Phi\Psi-\Psi\Phi} . $$

Since the normality of Φ is a necessary condition for the hyponormality of $T_{\Phi}$ (cf. [15]), the positivity of $H_{\Phi^{*}}^{*} H_{\Phi^{*}} - H_{\Phi}^{*}H_{\Phi}$ is an essential condition for the hyponormality of $T_{\Phi}$. If $\Phi\in L^{\infty}_{M_{n}}$, the pseudo-self-commutator of $T_{\Phi}$ is defined by

$$\bigl[T_{\Phi}^{*}, T_{\Phi}\bigr]_{p} := H_{\Phi^{*}}^{*} H_{\Phi^{*}} - H_{\Phi}^{*}H_{\Phi}. $$

Then $T_{\Phi}$ is said to be pseudo-hyponormal if $[T_{\Phi}^{*}, T_{\Phi}]_{p}\ge0$. We also see that if $\Phi\in L^{\infty}_{M_{n}}$ then $[T_{\Phi}^{*}, T_{\Phi}]= [T_{\Phi}^{*}, T_{\Phi}]_{p} + T_{\Phi^{*}\Phi-\Phi\Phi^{*}}$.

Proposition 2.2

Let $\Phi\equiv\Phi_{-}^{*} + \Phi_{+} \in L^{\infty}_{M_{n}}$ be such that Φ and $\Phi^{*}$ are of bounded type. Thus in view of (1.4), we may write

$$\Phi_{+}= \theta_{1} A^{*} \quad\textit{and}\quad \Phi_{-} = \theta_{2} B^{*}, $$

where $\theta_{1}$ and $\theta_{2}$ are inner functions and $A,B\in H^{2}_{M_{n}}$. If $T_{\Phi}$ is hyponormal then $\theta_{2}$ is an inner divisor of $\theta_{1}$, i.e., $\theta _{1}=\theta_{0} \theta_{2}$ for some inner function $\theta_{0}$.

Proof

See [7], Proposition 3.2. □

In view of Proposition 2.2, when we study the hyponormality of block Toeplitz operators with bounded type symbols Φ (i.e., Φ and $\Phi^{*}$ are of bounded type) we may assume that the symbol $\Phi\equiv\Phi_{-}^{*} + \Phi_{+}\in L^{\infty}_{M_{n}}$ is of the form

$$\Phi_{+}= \theta_{0} \theta_{1} A^{*} \quad\mbox{and}\quad \Phi_{-}=\theta_{0} B^{*}, $$

where $\theta_{0}$ and $\theta_{1}$ are inner functions and $A,B\in H^{2}_{M_{n}}$.

We first observe that if $\mathbf{T}=(T_{\varphi}, T_{\psi})$ then the self-commutator of T can be expressed as

$$ \bigl[\mathbf{T}^{*}, \mathbf{T}\bigr]= \begin{bmatrix} [T_{\varphi}^{*}, T_{\varphi}]& [T_{\psi}^{*}, T_{\varphi}]\\ [T_{\varphi}^{*}, T_{\psi}]& [T_{\psi}^{*}, T_{\psi}] \end{bmatrix} = \begin{bmatrix} H_{\overline{\varphi_{+}}}^{*} H_{\overline{\varphi_{+}}}-H_{\overline{\varphi_{-}}}^{*} H_{\overline{\varphi_{-}}} &H_{\overline{\varphi_{+}}}^{*} H_{\overline{\psi_{+}}} -H_{\overline{\psi_{-}}}^{*} H_{\overline{\varphi_{-}}}\\ H_{\overline{\psi_{+}}}^{*} H_{\overline{\varphi_{+}}}-H_{\overline{\varphi_{-}}}^{*} H_{\overline{\psi_{-}}} &H_{\overline{\psi_{+}}}^{*} H_{\overline{\psi_{+}}}-H_{\overline{\psi_{-}}}^{*} H_{\overline{\psi_{-}}} \end{bmatrix} . $$

(2.9)

For a block Toeplitz pair $\mathbf{T} \equiv(T_{\Phi}, T_{\Psi})$, the pseudo-commutator of T is defined by

$$\begin{aligned} \bigl[\mathbf{T}^{*}, \mathbf{T}\bigr]_{p} &:= \begin{bmatrix} [T_{\Phi}^{*}, T_{\Phi}]_{p} & [T_{\Psi}^{*}, T_{\Phi}]_{p}\\ [T_{\Phi}^{*}, T_{\Psi}]_{p} & [T_{\Psi}^{*}, T_{\Psi}]_{p} \end{bmatrix} \\ &= \begin{bmatrix} H_{\Phi_{+}^{*}}^{*} H_{\Phi_{+}^{*}}-H_{\Phi_{-}^{*}}^{*} H_{\Phi_{-}^{*}} & H_{\Phi_{+}^{*}}^{*} H_{\Psi_{+}^{*}}-H_{\Psi_{-}^{*}}^{*} H_{\Phi_{-}^{*}}\\ H_{\Psi_{+}^{*}}^{*} H_{\Phi_{+}^{*}}-H_{\Phi_{-}^{*}}^{*} H_{\Psi_{-}^{*}} & H_{\Psi_{+}^{*}}^{*} H_{\Psi_{+}^{*}}-H_{\Psi_{-}^{*}}^{*} H_{\Psi_{-}^{*}} \end{bmatrix} . \end{aligned}$$

Let $\Phi_{i} \in L^{\infty}_{M_{n}}$ ($i=1,2,\ldots,m$) be normal and mutually commuting and let σ be a permutation on $\{1,2,\ldots, m\}$. Then evidently,

$$ \begin{aligned}[b] &\mathbf {T}:=(T_{\Phi_{1}}, \ldots, T_{\Phi_{m}}) \mbox{ is hyponormal} \\ &\quad\Longleftrightarrow\quad \mathbf {T}_{\sigma}:=(T_{\Phi_{\sigma(1)}}, \ldots, T_{\Phi_{\sigma(m)}}) \mbox{ is hyponormal}. \end{aligned} $$

(2.10)

Moreover, we have

$$ \operatorname{rank} \bigl[\mathbf {T}^{*}, \mathbf {T}\bigr]= \operatorname{rank} \bigl[\mathbf {T}_{\sigma}^{*}, \mathbf {T}_{\sigma} \bigr]. $$

(2.11)

For every $m_{0} \leq m$, let $\mathbf {T}_{m_{0}}:=(T_{\Phi_{1}}, \ldots, T_{\Phi_{m_{0}}})$. Since

$$\bigl[\mathbf {T}^{*}, \mathbf {T}\bigr]= \begin{bmatrix} [\mathbf {T}_{\Phi_{m_{0}}}^{*}, \mathbf {T}_{\Phi_{m_{0}}}]& \ast\\ \ast&\ast \end{bmatrix} , $$

we can see that if T is hyponormal then in view of (2.10), every sub-tuple of T is hyponormal.

We then have the following.

Lemma 2.3

Let $\Phi_{i}\in L^{\infty}_{M_{n}}$ be normal and mutually commuting. Let $\mathbf {T}\equiv(T_{\Phi_{1}}, \ldots, T_{\Phi_{m}})$ and $\mathbf {S}\equiv(T_{\Lambda_{1}\Phi_{1}}, \ldots, T_{\Lambda_{m} \Phi_{m}})$, where the $\Lambda_{i}$ are mutually commuting and are invertible constant normal matrices commuting with $\Phi_{j}$ and $\Lambda_{j}$ for each $i,j=1,2,\ldots, m$. Then

$$\mathbf {T} \textit{ is hyponormal}\quad \Longleftrightarrow\quad \mathbf {S} \textit{ is hyponormal}. $$

Furthermore, $\operatorname{rank} [\mathbf {T}^{*}, \mathbf {T}]=\operatorname{rank} [\mathbf {S}^{*}, \mathbf {S}]$.

Proof

In view of equation (2.10), it suffices to prove the lemma when $\Lambda_{i}=I$ for all $i=2,\ldots, m$. Put $\mathcal {T}:=[\mathbf {T}^{*}, \mathbf {T}]$ and $\mathcal {S}:=[\mathbf {S}^{*}, \mathbf {S}]$. Since $\Lambda_{1}$ is a constant normal matrix commuting with $\Phi_{j}$, it follows that, for all $j >1$,

$$\begin{aligned} \mathcal {S}_{1j} &=H_{(\Lambda_{1} \Phi_{1})_{+}^{*}}^{*}H_{(\Phi_{j})_{+}^{*}} -H_{(\Phi_{j})_{-}^{*}}^{*}H_{(\Lambda_{1} \Phi_{1})_{-}^{*}} \\ &=H_{(\Phi_{1})_{+}^{*} \Lambda_{1}^{*}}^{*}H_{(\Phi_{j})_{+}^{*}}-H_{(\Phi _{j})_{-}^{*}}^{*}H_{\Lambda_{1} (\Phi_{1})_{-}^{*}} \\ &=T_{ \Lambda_{1}}H_{(\Phi_{1})_{+}^{*}}^{*}H_{(\Phi_{j})_{+}^{*}}-H_{(\Phi _{j})_{-}^{*}}^{*}T_{\Lambda_{1}}H_{(\Phi_{1})_{-}^{*}} \\ &=T_{ \Lambda_{1}}H_{(\Phi_{1})_{+}^{*}}^{*} H_{(\Phi_{j})_{+}^{*}} - H_{(\Phi _{j})_{-}^{*}\Lambda_{1}^{*}}^{*} H_{(\Phi_{1})_{-}^{*}} \\ &=T_{ \Lambda_{1}} \bigl(H_{(\Phi_{1})_{+}^{*}}^{*}H_{(\Phi_{j})_{+}^{*}} - H_{(\Phi_{j})_{-}^{*}}^{*} H_{(\Phi_{1})_{-}^{*}} \bigr) \\ &=T_{ \Lambda_{1}}\mathcal {T}_{1j}. \end{aligned}$$

Observe that

$$\begin{aligned} \mathcal {S}_{11} &=H_{(\Lambda_{1} \Phi_{1})_{+}^{*}}^{*}H_{(\Lambda_{1} \Phi_{1})_{+}^{*}} -H_{(\Lambda_{1} \Phi_{1})_{-}^{*}}^{*}H_{(\Lambda_{1} \Phi_{1})_{-}^{*}} \\ &=H_{(\Phi_{1})_{+}^{*} \Lambda_{1}^{*}}^{*} H_{(\Phi_{1})_{+}^{*} \Lambda_{1}^{*}} - H_{(\Phi_{1})_{-}^{*} \Lambda_{1}^{*}}^{*}H_{(\Phi_{1})_{-}^{*} \Lambda_{1}^{*}} \\ &=T_{ \Lambda_{1}}H_{(\Phi_{1})_{+}^{*}}^{*}H_{(\Phi_{1})_{+}^{*}}T_{\Lambda_{1}}^{*} - T_{ \Lambda_{1}} H_{(\Phi_{1})_{-}^{*}}^{*}H_{(\Phi_{1})_{-}^{*}}T_{\Lambda _{1}}^{*} \\ &=T_{ \Lambda_{1}} \bigl(H_{(\Phi_{1})_{+}^{*}}^{*}H_{(\Phi_{1})_{+}^{*}} -H_{(\Phi_{1})_{-}^{*}}^{*}H_{(\Phi_{1})_{-}^{*}} \bigr)T_{\Lambda_{1}}^{*} \\ &=T_{ \Lambda_{1}}\mathcal {T}_{11}T_{ \Lambda_{1}}^{*}. \end{aligned}$$

Let Q be the block diagonal operator with the diagonal entries $(T_{\Lambda_{1}}, I, \ldots, I)$. Then Q is invertible and $\mathcal {S}=Q\mathcal {T} Q^{*}$, which gives the result. □

Lemma 2.4

Let $\mathbf {T}\equiv(T_{\Phi_{1}}, T_{\Phi_{2}}, \ldots T_{\Phi_{m}})$, where the $\Phi_{i}\in L^{\infty}_{M_{n}}$ ($i=1,\ldots,m$) are normal and mutually commuting. If $\mathbf {S}:=(T_{\Phi_{1}-\Phi_{j_{0}}}, T_{\Phi_{2}}, \ldots T_{\Phi_{m}})$ for some $j_{0}$ ($2\le j_{0}\le m$), then

$$\mathbf {T} \textit{ is hyponormal}\quad \Longleftrightarrow\quad \mathbf {S} \textit{ is hyponormal}. $$

Furthermore, $\operatorname{rank} [\mathbf {T}^{*}, \mathbf {T}]=\operatorname{rank} [\mathbf {S}^{*}, \mathbf {S}]$.

Proof

Obvious. □

Corollary 2.5

Let $\Phi_{i}\in L^{\infty}_{M_{n}}$ ($i=1,\ldots,m$) be normal and mutually commuting. Let $\mathbf {T}\equiv(T_{\Phi_{1}}, \ldots T_{\Phi_{m}})$ and put

$$\mathbf {S}:=(T_{\Phi_{1}-\Lambda_{1}\Phi_{m}}, T_{\Phi_{2}-\Lambda_{2}\Phi_{m}},\ldots, T_{\Phi_{m-1}-\Lambda_{m-1} \Phi_{m}}, T_{\Phi_{m}}), $$

where the $\Lambda_{i}$ ($i=1,\ldots,m-1$) are mutually commuting and are invertible constant normal matrices commuting with $\Phi_{j}$ for each $j=1,\ldots, m$. Then

$$\mathbf {T} \textit{ is hyponormal} \quad\Longleftrightarrow\quad \mathbf {S} \textit{ is hyponormal}. $$

Furthermore, $\operatorname{rank} [\mathbf {T}^{*}, \mathbf {T}]=\operatorname{rank} [\mathbf {S}^{*}, \mathbf {S}]$.

Proof

This follows from Lemmas 2.3 and 2.4. □

We now have the following.

Theorem 2.6

Let $\Phi_{i}\in H^{\infty}_{M_{n}}$ ($i=1,2,\ldots,m-1$) be mutually commuting and normal rational functions of the form

$$\Phi_{i} = A_{i}^{*} \Theta_{i} \quad(\textit{left coprime}), $$

where the $\Theta_{i}$ are inner matrix functions and $\Phi_{m} \equiv (\Phi_{m})_{-}^{*} + (\Phi_{m})_{+} \in L^{\infty}_{M_{n}}$. If $\mathbf{T}:=(T_{\Phi_{1}},\ldots, T_{\Phi_{m}})$ is hyponormal then

$$ \operatorname{rank} \bigl[\mathbf {T}^{*}, \mathbf {T}\bigr] = \operatorname{deg} (\Theta)+\operatorname{rank} \bigl[T_{\Phi_{m}^{1,\Theta}}^{*}, T_{\Phi_{m}^{1,\Theta}}\bigr]_{p}, $$

(2.12)

where $\Theta:=\textit{right-l.c.m.} \{\Theta_{i}: i=1,2,\ldots ,m-1\}$ and $\Phi_{m}^{1,\Theta}:= (\Phi_{m})_{-}^{*}+P_{H_{0}^{2}}((\Phi_{m})_{+}\Theta^{*})$.

Proof

Let $\mathbf{H}_{\Phi^{*}}:=(H_{\Phi_{1}^{*}},\ldots, H_{\Phi_{m-1}^{*}})$. Since $\Phi_{i}\equiv(\Phi_{i})_{+} \in H^{\infty}_{M_{n}} (i=1,2,\ldots,m-1)$, T is hyponormal if and only if

$$\bigl[\mathbf{T}^{*}, \mathbf{T}\bigr]= \begin{bmatrix} \mathbf{H}_{\Phi^{*}}^{*}\mathbf{H}_{\Phi^{*}}&\mathbf{H}_{\Phi ^{*}}^{*}H_{\Phi_{m}^{*}}\\ H_{\Phi_{m}^{*}}^{*}\mathbf{H}_{\Phi^{*}}&[T_{\Phi_{m}}^{*}, T_{\Phi_{m}}] \end{bmatrix} \geq0, $$

or equivalently, for each $X \in\bigoplus_{j=1}^{m-1} H^{2}_{\mathbb{C}^{n}}$ and $Y \in H^{2}_{\mathbb{C}^{n}}$,

$$ \bigl\vert \bigl\langle \mathbf{H}_{\Phi^{*}}H_{\Phi_{m}^{*}}^{*} Y, X \bigr\rangle \bigr\vert ^{2} \leq \bigl\langle \mathbf{H}_{\Phi^{*}}^{*} \mathbf{H}_{\Phi^{*}} X, X \bigr\rangle \bigl\langle \bigl[T_{\Phi_{m}}^{*}, T_{\Phi_{m}}\bigr] Y , Y \bigr\rangle . $$

(2.13)

Since $\operatorname{cl\,ran} H_{\Phi_{i}^{*}}=\mathcal {H}(\widetilde{\Theta}_{i})$ ($i=1,2,\ldots, n-1$), it follows that

$$ \begin{aligned}[b] \operatorname{cl\,ran} \mathbf{H}_{\Phi^{*}} &=\bigvee_{i=1}^{m-1} \operatorname{cl\,ran} H_{\Phi_{i}^{*}} =\bigvee_{i=1}^{m-1} \mathcal {H} (\widetilde{\Theta}_{i}) = \Biggl(\bigcap _{i=1}^{m-1}\widetilde{\Theta}_{i} H_{\mathbb {C}^{n}}^{2} \Biggr)^{\perp} \\ &= \bigl(\widetilde{\Theta} H_{\mathbb {C}^{n}}^{2} \bigr)^{\perp} =\mathcal {H} (\widetilde{\Theta}) =\operatorname{cl\,ran}H_{\Theta^{*}}, \end{aligned} $$

(2.14)

where $\mathcal{H}(\Delta):=H^{2}_{\mathbb {C}^{n}}\ominus\Delta H^{2}_{\mathbb {C}^{n}}$. If the $\Phi_{i}$ are rational functions then, by (1.3) and (1.4), we can write

$$\Phi_{i}=\theta_{i} A_{i}^{*} \quad ( \theta_{i}, \mbox{ finite Blaschke product}). $$

Since $\Theta_{i}$ is a right inner divisor of $I_{\theta_{i}}$, we have $\operatorname{deg}(\Theta_{i}) \leq\operatorname{deg}(I_{\theta_{i}})=n \operatorname{deg}(\theta_{i})<\infty$. Thus since by (2.14), $\operatorname{cl\,ran} \mathbf{H}_{\Phi^{*}}=\mathcal {H} (\widetilde{\Theta})$ and

$$\operatorname{deg}(\Theta)=\operatorname{rank} H_{\Theta^{*}}^{*}= \operatorname{rank} H_{\Theta^{*}}=\operatorname{deg} (\widetilde{\Theta})< \infty. $$

Therefore $\mathbf{H}_{\Phi^{*}}$ is of finite rank and hence, so is $\mathbf{H}_{\Phi^{*}}^{*}\mathbf{H}_{\Phi^{*}}$ and, moreover,

$$\operatorname{rank} \bigl(\mathbf{H}_{\Phi^{*}}^{*}\mathbf{H}_{\Phi^{*}} \bigr) = \operatorname{rank} \bigl(\mathbf{H}_{\Phi^{*}}^{*}\bigr) = \operatorname{rank} (\mathbf{H}_{\Phi^{*}}) = \operatorname{deg} (\Theta) . $$

Thus by (2.7), we have

$$\begin{aligned} \operatorname{rank} \bigl[\mathbf{T}^{*}, \mathbf{T}\bigr] &=\operatorname{rank} \begin{bmatrix} \mathbf{H}_{\Phi^{*}}^{*}\mathbf{H}_{\Phi^{*}} &\mathbf{H}_{\Phi^{*}}^{*}H_{\Phi_{m}^{*}}\\ H_{\Phi_{m}^{*}}^{*}\mathbf{H}_{\Phi^{*}} &[T_{\Phi_{m}}^{*}, T_{\Phi_{m}}] \end{bmatrix} \\ & =\operatorname{rank} \bigl(\mathbf{H}_{\Phi^{*}}^{*}\mathbf{H}_{\Phi ^{*}} \bigr)+\operatorname{rank} \bigl( \bigl[T_{\Phi_{m}}^{*}, T_{\Phi_{m}} \bigr]-H_{\Phi_{m}^{*}}^{*}\mathbf{H}_{\Phi^{*}} \bigl(\mathbf{H}_{\Phi^{*}}^{*} \mathbf{H}_{\Phi^{*}}\bigr)^{\sharp}\mathbf {H}_{\Phi^{*}}^{*}H_{\Phi_{m}^{*}} \bigr) \\ & =\operatorname{deg}(\Theta)+\operatorname{rank} \bigl( \bigl[T_{\Phi_{m}}^{*}, T_{\Phi_{m}}\bigr]-H_{\Phi_{m}^{*}}^{*}\mathbf{H}_{\Phi^{*}}\bigl( \mathbf{H}_{\Phi^{*}}^{*} \mathbf{H}_{\Phi^{*}}\bigr)^{\sharp} \mathbf{H}_{\Phi^{*}}^{*}H_{\Phi _{m}^{*}} \bigr). \end{aligned}$$

On the other hand, by Proposition 2.1, $\mathbf{H}_{\Phi^{*}}(\mathbf{H}_{\Phi^{*}}^{*} \mathbf{H}_{\Phi^{*}})^{\sharp}\mathbf{H}_{\Phi^{*}}^{*}$ is the projection $P_{\mathcal {H} (\widetilde{\Theta})}$. Therefore it follows from (1.7) and (1.8) that

$$\begin{aligned} &\bigl[T_{\Phi_{m}}^{*}, T_{\Phi_{m}}\bigr] -H_{\Phi_{m}^{*}}^{*} \mathbf{H}_{\Phi^{*}} \bigl(\mathbf{H}_{\Phi^{*}}^{*}\mathbf{H}_{\Phi^{*}} \bigr)^{\sharp}\mathbf {H}_{\Phi^{*}}^{*}H_{\Phi_{m}^{*}} \\ &\quad=\bigl[T_{\Phi_{m}}^{*}, T_{\Phi_{m}}\bigr]-H_{\Phi_{m}^{*}}^{*}H_{\Theta^{*}}H_{\Theta ^{*}}^{*}H_{\Phi_{m}^{*}} \\ &\quad=H_{\Phi_{m+}^{*}}^{*}\bigl(I-H_{\Theta^{*}}H_{\Theta^{*}}^{*} \bigr)H_{\Phi _{m+}^{*}}-H_{\Phi_{m-}^{*}}^{*}H_{\Phi_{m-}^{*}} \\ &\quad=\bigl(H_{\Phi_{m+}^{*}}^{*}T_{\widetilde{\Theta}}\bigr) (T_{\widetilde{\Theta }^{*}}H_{\Phi_{m+}^{*}}) -H_{\Phi_{m-}^{*}}^{*}H_{\Phi_{m-}^{*}} \\ &\quad=H_{\Theta\Phi_{m+}^{*}}^{*}H_{\Theta\Phi_{m+}^{*}}-H_{\Phi _{m-}^{*}}^{*}H_{\Phi_{m-}^{*}} \\ &\quad=\bigl[T_{\Phi_{m}^{1,\Theta}}^{*}, T_{\Phi_{m}^{1,\Theta}}\bigr]_{p}, \end{aligned}$$

which gives the result. □

Very recently, the hyponormality of rational Toeplitz pairs was characterized in [16].

Lemma 2.7

(Hyponormality of rational Toeplitz pairs) [16]

Let $\mathbf{T}\equiv(T_{\Phi}, T_{\Psi})$ be a Toeplitz pair with rational symbols $\Phi, \Psi\in L^{\infty}_{M_{n}}$ of the form

$$ \Phi_{+} = \theta_{0} \theta_{1} A^{*}, \qquad \Phi_{-} =\theta_{0} B^{*}, \qquad \Psi_{+} = \theta_{2} \theta_{3} C^{*}, \qquad \Psi_{-} = \theta_{2} D^{*} \quad( \textit{coprime}). $$

(2.15)

Assume that $\theta_{0}$ and $\theta_{2}$ are not coprime. Assume also that $B(\gamma_{0})$ and $D(\gamma_{0})$ are diagonal-constant for some $\gamma_{0}\in\mathcal{Z}(\theta_{0})$. Then the pair T is hyponormal if and only if

• (i) Φ and Ψ are normal and $\Phi\Psi=\Psi\Phi$;

• (ii) $\Phi_{-}=\Lambda^{*}\Psi_{-}$ (with $\Lambda:=B(\gamma _{0})D(\gamma_{0})^{-1}$);

• (iii) $T_{\Psi^{1, \Omega}}$ is pseudo-hyponormal with $\Omega:=\theta_{0}\theta_{1}\theta_{3}\overline{\theta}\Delta^{*}$,

where $\theta:=\textit{g.c.d.} (\theta_{1}, \theta_{3})$ and $\Delta:=\textit{left-g.c.d.} (I_{\theta_{0}\theta}, \overline{\theta}(\theta_{3}A- \theta_{1} C\Lambda^{*}) )$.

We now get a rank formula for the self-commutators of Toeplitz m-tuples.

Corollary 2.8

For each $i=1,2,\ldots, m$, suppose that $\Phi_{i}=(\Phi_{i})_{-}^{*}+(\Phi_{i})_{+} \in L^{\infty}_{M_{n}}$ is a matrix-valued normal rational function of the form

$$(\Phi_{i})_{+} = \theta_{i} \delta_{i} A_{i}^{*} \quad\textit{and}\quad (\Phi_{i})_{-} = \theta_{i} B_{i}^{*} \quad(\textit{coprime}), $$

where the $\theta_{i}$ and the $\delta_{i}$ are finite Blaschke products and there exists $j_{0}$ ($1\le j_{0}\le m$) such that $\theta_{j_{0}}$ and $\theta_{i}$ are not coprime for each $i=1,2,\ldots,m$. Suppose $\Phi_{i}\Phi_{j}=\Phi_{j}\Phi_{i}$ for all $i,j=1,\ldots,m$. Assume that each $B_{i}(\gamma_{0})$ is diagonal-constant for some $\gamma_{0}\in\mathcal{Z}(\theta_{i})$. If $\mathbf {T}\equiv(T_{\Phi_{1}}, T_{\Phi_{2}},\ldots, T_{\Phi_{m}})$ is hyponormal then

$$\operatorname{rank} \bigl[\mathbf {T}^{*}, \mathbf {T}\bigr] =\operatorname{deg} ( \Omega)+\operatorname{rank} \bigl[T_{\Phi_{j_{0}}^{1,\Omega}}^{*}, T_{\Phi_{j_{0}}^{1,\Omega}} \bigr]_{p}, $$

where $\Omega:=\textit{right-l.c.m.} \{\theta_{i}\delta_{i}\delta_{j_{0}} \overline{\delta(i)} \Theta(i)^{*}: i=1,2, \ldots, m\}$. Here $\delta(i):=\textit{g.c.d.} \{\delta_{i}, \delta_{j_{0}}\}$ and $\Theta(i):=\textit{left-g.c.d.} \{\theta_{i}\delta(i), \overline{\delta(i)} (\delta_{j_{0}}A_{i}- \delta_{i} A_{j_{0}}\Lambda (i)^{*} )\}$ with $\Lambda(i):=B_{i}(\gamma_{0})B_{j_{0}}(\gamma_{0})^{-1}$.

Proof

Suppose T is hyponormal. Since every sub-tuple of T is hyponormal, we can see that $(T_{\Phi_{i}}, T_{\Phi_{j}})$ is hyponormal for all $i,j=1,2, \ldots,m$. In view of (2.10), we may assume that $j_{0}=m$. Put

$$\mathbf {S}:=(T_{\Phi_{1}-\Lambda(1)\Phi_{m}}, T_{\Phi_{2}-\Lambda(2)\Phi_{m}}, \ldots, T_{\Phi_{m-1}-\Lambda(m-1) \Phi_{m}}, T_{\Phi_{m}}). $$

It follows from Corollary 2.5 that

$$\mathbf {T} \mbox{ is hyponormal} \quad\Longleftrightarrow\quad \mathbf {S} \mbox{ is hyponormal}. $$

Since $\delta(i)=\mbox{g.c.d.}\{\delta_{i}, \delta_{m}\}$, we can write

$$\delta_{i}=\delta(i) \omega_{i} \quad \mbox{and}\quad \delta_{m}=\delta(i) \omega_{m} , $$

where $\omega_{i}$ is a finite Blaschke product for $i=1,2,\ldots,m$. Since $\Theta(i)=\mbox{left-g.c.d.} \{\theta_{i}\delta(i), \overline{\delta(i)} (\delta_{m} A_{i} - \delta_{1} A_{m} \Lambda(i)^{*} ) \}$, we get the following left coprime factorization:

$$\Phi_{i}-\Lambda(i)\Phi_{m}= \bigl[\bigl(\overline{ \omega_{m}} A_{i}^{*}- \overline{\omega_{i}} \Lambda(i) A_{m}^{*}\bigr)\Theta(i) \bigr]\theta_{i} \delta_{i}\delta_{m} \overline{\delta (i)}\Theta(i)^{*} . $$

Thus the result follows at once from Theorem 2.6. □

We conclude with the following.

Corollary 2.9

For each $i=1,2,\ldots, m$, suppose that $\phi_{i}=\overline{(\phi_{i})_{-}}+(\phi_{i})_{+} \in L^{\infty}$ is a rational function of the form

$$(\phi_{i})_{+} = \theta_{i}\overline{a_{i}} \quad \textit{and}\quad (\phi_{i})_{-} =\theta_{i} \overline{b_{i}} \quad(\textit{coprime}). $$

If there exists $j_{0}$ ($1\le j_{0}\le m$) such that $\theta_{j_{0}}$ and $\theta_{i}$ are not coprime for each $i=1,2,\ldots,m$ and $\mathbf {T}\equiv(T_{\phi_{1}}, T_{\phi_{2}},\ldots, T_{\phi_{m}})$ is hyponormal then

$$\operatorname{rank} \bigl[\mathbf {T}^{*}, \mathbf {T}\bigr] =\operatorname{rank} \bigl[T_{\Phi_{j_{0}}}^{*}, T_{\Phi_{j_{0}}} \bigr]. $$

Proof

For each $i=1,2,\ldots, m$, let $\lambda(i):=b_{i}(\gamma_{0})b_{j_{0}}(\gamma_{0})^{-1}$ for some $\gamma_{0}\in\mathcal{Z}(\theta_{i})$. Write $\theta(i)\equiv\mbox{g.c.d.} \{\theta_{i}, (a_{i}- a_{j_{0}}\overline{\lambda(i)} ) \}$. Since $\mathbf {T}\equiv(T_{\phi_{1}}, T_{\phi_{2}},\ldots, T_{\phi_{n}})$ is hyponormal, $(T_{\phi_{i}}, T_{\phi_{j_{0}}})$ is hyponormal for all $i=1,2,\ldots,n$. Thus it follows from Lemma 2.7 that $T_{\phi_{j_{0}}^{1,\omega(i)}}$ is hyponormal with $\omega(i):=\theta_{i} \overline{\theta(i)}$. Observe that

$$\bigl(\phi_{j_{0}}^{1,\omega(i)}\bigr)_{+} = \theta(i) \overline{c_{i}} \quad\mbox{and} \quad\bigl(\phi_{j_{0}}^{1, \omega(i)} \bigr)_{-} =\theta_{i} \overline{b_{i}} \quad(\mbox{coprime}), $$

where $c_{i}:=P_{\mathcal {H}( \theta(i))}(a_{i})$. Since $T_{\phi_{j_{0}}^{1,\omega(i)}}$ is hyponormal, it follows from Proposition 2.2 that $\theta_{i}$ is an inner divisor of $\theta(i)$ and hence $\theta(i)=\theta_{i}$. Thus the result follows from Corollary 2.8. □

3 Conclusions

The self-commutators of bounded linear operators play an important role in the study of hyponormal and subnormal operators. The main result of this paper is to derive a rank formula for the self-commutators of tuples of Toeplitz operators with matrix-valued rational symbols. This result will contribute to the study of Toeplitz operators and the bridge theory of operators.