Hyponormal block Toeplitz operators with finite rank self-commutators

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 $[T_\varphi^{\ast}, T_\varphi]$ 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 [10, Conjecture 6.1]. 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 ; $\breve{\Phi}(e^{i\theta}):=\Phi(e^{-i\theta})$.