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 $T_φ$ is hyponormal and $[T_φ^{*}, T_φ]$ is a finite rank operator if and only if there exists a finite Blaschke product $b$ in $\mathcal{E}(φ)$, where $$ \mathcal{E}(φ) := \{k \in H^\infty(\mathbb{T}): \left\|k\right\|_\infty \le 1 \textrm{ and } φ-k\cdot \barφ \in H^\infty(\mathbb{T})\}. $$ An analogous set $\mathcal{E}(Φ)$ can be defined for a matrix-valued symbol $Φ$. \ In the block Toeplitz operator case, we first establish that if a symbol $Φ$ is in $L^\infty(\mathbb{T}, M_n)$ and if $\mathcal{E}(Φ)$ contains a constant unitary matrix $U$, then $T_Φ$ 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 $Φ\in H^{\infty}(\mathbb{T}, M_n)$ is such that $Φ^{\ast}$ is of bounded type and $T_Φ$ is hyponormal. \ Then $[T_Φ^{\ast}, T_Φ]$ is a finite rank operator if and only if there exists a finite Blaschke-Potapov product in $\mathcal{E}(\widetildeΦ)$, where $\widetildeΦ:=\breveΦ^*$ and $\breveΦ(e^{iθ}):=Φ(e^{-iθ})$.