Open Access

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

Journal of Inequalities and Applications20162016:184

https://doi.org/10.1186/s13660-016-1125-x

Received: 14 April 2016

Accepted: 2 July 2016

Published: 22 July 2016

Abstract

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

Keywords

block Toeplitz operators jointly hyponormal bounded type functions rational functions self-commutators

MSC

47B20 47B35 47A13 30H10 47A57

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 [14].

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 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 \(\theta_{i} \) (\(i \in J\)).
  1. (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 \(\theta_{d}=\textit{g.c.d.} \{\theta_{i} : i \in J \}\).

     
  2. (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
  1. (i)

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

     
  2. (ii)

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

     
  3. (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.

Declarations

Acknowledgements

The work of the first author was supported by National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science and Technology (No. 2011-0022577). The work of the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015R1D1A3A01016258).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Sungkyunkwan University
(2)
Department of Mathematics, Changwon National University

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