Subnormal block Toeplitz operators

Abstract

In this paper we consider the subnormality of block Toeplitz operators $T_\Phi$, where $\Phi$ is an $n\times n$ matrix-valued function on the unit circle $\mathbb T$ of the form $$ \Phi=Q\Phi^* \quad \hbox{($Q$ is a finite Blaschke--Potapov product).} $$ This is related to a matrix-valued version of Halmos's Problem 5 and Nakazi-Takahashi Theorem. We ask whether $T_\Phi$ is either normal or analytic if $T_\Phi$ is subnormal, where $\Phi$ is of the above form. We give answers to this problem for different cases of the symbol. Moreover, we provide a sufficient condition for the answer to be affirmative when $\Phi^*$ is not of bounded type.