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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 \(\theta_{i} \) (\(i \in J\)).

  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.

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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).

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Correspondence to An Hyun Kim.

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Hwang, I.S., Kim, A.H. & Kim, S. A rank formula for the self-commutators of rational Toeplitz tuples. J Inequal Appl 2016, 184 (2016). https://doi.org/10.1186/s13660-016-1125-x

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