Commutative algebras of Toeplitz operators on the Bergman space revisited: Spectral theorem approach

TL;DR

This paper revisits commutative Toeplitz operator algebras on Bergman spaces, providing spectral representations and analyzing the relationship between spectral functions and operator symbols.

Contribution

It introduces spectral theorem-based representations for these algebras, linking spectral functions to Toeplitz operator symbols in Bergman spaces.

Findings

Representation of Toeplitz algebras as bounded functions of unbounded self-adjoint operators

Relations between spectral functions and operator symbols

Properties of spectral functions in the spectral representation

Abstract

For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols.