Semi-commutants of Toeplitz Operators on Fock-Sobolev space of Nonnegative orders

TL;DR

This paper investigates the semi-commutants of Toeplitz operators on Fock-Sobolev spaces, revealing fundamental differences from classical Fock spaces and providing new insights into their operator algebra structure.

Contribution

It generalizes previous results to Fock-Sobolev spaces and characterizes when Toeplitz operators semi-commute, highlighting key differences from classical Fock space behavior.

Findings

Semi-commuting Toeplitz operators are trivial in certain symbol spaces.

Differences between Fock and Fock-Sobolev space geometries are fundamental.

Results challenge previous conjectures based on classical Fock space analysis.

Abstract

We make a progress towards describing the semi-commutants of Toeplitz operators on Fock-Sobolev spaces of nonnegative orders. We generalize the results in \cite{Bauer1,Qin}. For the certain symbol spaces, we obtain two Toeplitz operators can semi-commute only in the trivial cases, which is different from what is known for the classical Fock spaces. As an application, we consider the conjecture which was shown to be false for Fock space in \cite{MA}. The main results of this paper say that there is the fundamental difference between the geometries of Fock and Fock-Sobolev space.