Subnormal block Toeplitz operators

TL;DR

This paper investigates the subnormality of block Toeplitz operators with matrix-valued symbols involving Blaschke–Potapov products, addressing a matrix version of classical problems and providing conditions for normality or analyticity.

Contribution

It extends the understanding of subnormal block Toeplitz operators with specific matrix-valued symbols, offering new criteria and solutions for related classical problems.

Findings

Subnormal block Toeplitz operators are either normal or analytic under certain conditions.

A sufficient condition is provided when the symbol's adjoint is not of bounded type.

Answers are given for different cases of the symbol related to classical problems.

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.