Which subnormal Toeplitz operators are either normal or analytic?

TL;DR

This paper characterizes subnormal Toeplitz operators on vector-valued Hardy spaces, extending Abrahamse's Theorem to matrix symbols and exploring conditions under which they are either normal or analytic.

Contribution

It extends Abrahamse's Theorem to matrix-valued symbols and establishes the necessity of coprime decomposition conditions for subnormal Toeplitz operators.

Findings

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

The coprime decomposition condition is proven to be essential.

The paper examines the conjecture on subnormal Toeplitz operators with finite rank self-commutator.

Abstract

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos's Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse's Theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two analytic functions), whose co-analytic part has a "coprime decomposition," is normal or analytic. We also prove that the coprime decomposition condition is essential. Finally, we examine a well known conjecture, of whether every submormal Toeplitz operator with finite rank self-commutator is normal or analytic.