Finite Rank Perturbations of Toeplitz Products on the Bergman Space

Trieu Le; Damith Thilakarathna

doi:10.1016/j.jfa.2020.108850

paper

Finite Rank Perturbations of Toeplitz Products on the Bergman Space

arXiv:2011.05414 · doi:10.1016/j.jfa.2020.108850

Abstract

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution $⋄$ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if $F_{j}, G_{j}$ ( $1 \leq j \leq N$ ) are polynomials of $z$ and $z ˉ$ then $\sum_{j = 1}^{N} T_{F_{j}} T_{G_{j}} - T_{H}$ is a finite rank operator for some $L^{1}$ -function $H$ if and only if $\sum_{j = 1}^{N} F_{j} ⋄ G_{j}$ belongs to $L^{1}$ and $H = \sum_{j = 1}^{N} F_{j} ⋄ G_{j}$ . In the case $F_{j}$ 's are holomorphic and $G_{j}$ 's are conjugate holomorphic, it is shown that $H$ is a solution to a system of first order partial differential equations with a constraint.