Monomial-type Toeplitz operators on some weakly pseudoconvex domains

TL;DR

This paper fully characterizes the finite rank commutator and semi-commutator of monomial-type Toeplitz operators on certain weakly pseudoconvex domains, revealing new commuting and semi-commuting operators and extending known results.

Contribution

It provides a complete characterization of finite rank commutators and semi-commutators of monomial-type Toeplitz operators on specific weakly pseudoconvex domains, with novel higher-dimensional insights.

Findings

Existence of many commuting monomial-type Toeplitz operators.

Presence of non-trivial semi-commuting monomial-type Toeplitz operators.

Results extend known theories with new higher-dimensional phenomena.

Abstract

In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.%The situation is different from the case of unit disk. %Our results extend several known results using completely different arguments. Some interesting higher-dimensional phenomena appear on the unit polydisk.