Hyponormal block Toeplitz operators with finite rank self-commutators

Abstract

In this paper, we identify a large class of hyponormal block Toeplitz operators whose self-commutators are of finite rank. \ Recall that an operator $T_\varphi$ is hyponormal and $[T_\varphi^{*}, T_\varphi]$ is a finite rank operator if and only if there exists a finite Blaschke product $b$ in $\mathcal{E}(\varphi)$, where $$ \mathcal{E}(\varphi) := \{k \in H^\infty(\mathbb{T}): \left\|k\right\|_\infty \le 1 \textrm{ and } \varphi-k\cdot \bar{\varphi} \in H^\infty(\mathbb{T})\}. $$ An analogous set $\mathcal{E}(\Phi)$ can be defined for a matrix-valued symbol $\Phi$. \ In the block Toeplitz operator case, we first establish that if a symbol $\Phi$ is in $L^\infty(\mathbb{T}, M_n)$ and if $\mathcal{E}(\Phi)$ contains a constant unitary matrix $U$, then $T_\Phi$ is normal. \ We then obtain a suitable converse, under a mild assumption on the symbol. \ Next, we provide a partial answer to a conjecture recently posed by R.E. Curto, I.S. Hwang, and W.Y. Lee. \ Concretely, assume that $\Phi \in H^{\infty}(\mathbb{T}, M_n)$ is such that $\Phi^{\ast}$ is of bounded type and $T_\Phi$ is hyponormal. \ Then $[T_\Phi^{\ast}, T_\Phi]$ is a finite rank operator if and only if there exists a finite Blaschke-Potapov product in $\mathcal{E}(\widetilde{\Phi})$, where $\widetilde\Phi:=\breve{\Phi}^*$ and $\breve{\Phi}(e^{i\theta}):=\Phi(e^{-i\theta})$.