Centralizers of Toeplitz operators with polynomial symbols

TL;DR

This paper characterizes the centralizers of Toeplitz operators with polynomial symbols on the Bergman space, showing that elements commuting with nonconstant polynomial Toeplitz operators are bounded holomorphic Toeplitz operators.

Contribution

It provides a complete description of centralizers for polynomial-symbol Toeplitz operators on the Bergman space, extending understanding of their algebraic structure.

Findings

Centralizers of polynomial-symbol Toeplitz operators are characterized.

Elements commuting with nonconstant polynomial Toeplitz operators are bounded holomorphic Toeplitz operators.

The algebra generated by Toeplitz operators has a specific structure related to bounded holomorphic functions.

Abstract

In this note we describe centralizers of Toeplitz operators with polynomial symbols on the Bergman space. As a consequence it is shown that if an element of the norm closed algebra generated by all Toeplitz operators commutes with a Toeplitz operator of a noncon- stant polynomial, then this element is a Toeplitz operator of a bounded holomorphic function.