Which hyponormal block Toeplitz operators are either normal or analytic?

TL;DR

This paper investigates conditions under which hyponormal block Toeplitz operators are either normal or analytic, extending previous work to operators with bounded type symbols and specific inner function properties.

Contribution

It extends the characterization of hyponormal block Toeplitz operators to those with bounded type symbols and scalar inner functions, showing they are either normal or analytic.

Findings

Hyponormal block Toeplitz operators with certain bounded type symbols are either normal or analytic.

Extension of previous results to operators with symbols of bounded type and specific inner function conditions.

Hyponormality of the square of the operator implies it is either normal or analytic.

Abstract

In this paper, we continue Curto-Hwang-Lee's work to study the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle. Curto-Hwang-Lee's work focuses primarily on hyponormality and subnormality of block Toeplitz operators with rational symbols. By studying the greatest common divisor of matrix-valued inner functions and the ``weak" commutativity of matrix-valued inner functions, we extended Curto-Hwang-Lee's result to block Toeplitz operators with symbols of bounded type. More precisely, we proved that if $\Psi,\Psi^{\ast}$ are matrix-valued functions of bounded type and the inner part of $\Psi$ of Douglas-Shapiro-Shields factorization is a scalar inner function, then every hyponormal Toeplitz operator $T_{\Psi}$ whose square is also hyponormal must be either normal or analytic.