Commuting Slant Toeplitz Operators on the Bergman Space

TL;DR

The paper addresses when two slant Toeplitz operators on the Bergman space commute, focusing on symbols that share a conjugate-polynomial part. It models slant Toeplitz operators as $B_f=WT_f$ and analyzes products $B_fB_g$ versus $B_gB_f$ by decomposing symbols into a conjugate polynomial and analytic components. The main result shows that for $f=\overline{p}+\varphi$ and $g=\overline{p}+\psi$ with $p$ a polynomial and $\varphi,\psi$ bounded analytic, commutativity occurs if and only if $\varphi=\psi$, a Brown–Halmos-type theorem in this setting. This clarifies the commutant structure of slant Toeplitz operators on the Bergman space and has implications for spectral and operator-algebraic properties within this framework.

Abstract

This paper shows that on the Bergman space of the open unit disk, the slant Toeplitz operator $T_{p+\varphi}$ and $T_{p+ψ}$ commute if and only if $\varphi=ψ$ ,where $\varphi$ and $ψ$ are both bounded analytic functions, and $p$ is ananalytic polynomial.

Theorems & Definitions (12)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• proof
• Theorem 3.4