Toeplitz Operators on the Generalized Fock Space in Symmetrically Normed Ideals

TL;DR

This paper characterizes when Toeplitz operators on the generalized Fock space belong to symmetrically normed ideals, providing necessary and sufficient conditions based on the measure symbol and the operator parameter s.

Contribution

It offers a complete characterization of Toeplitz operators' membership in symmetrically normed ideals on the generalized Fock space, extending previous results to arbitrary symmetric norming functions.

Findings

Derived necessary and sufficient conditions for Toeplitz operators to be in symmetrically normed ideals.

Extended the analysis to arbitrary symmetric norming functions.

Provided a framework for understanding operator ideal membership in the generalized Fock space.

Abstract

In this paper we will give necessary and sufficient conditions for the operator $T_\nu^s$ to be in the symmetrically normed ideal $\mathcal{C}_\Phi$ for an arbitrary symmetric norming function $\Phi$ where $T_\nu$ is the Toeplitz operator on the generalized Fock Space (also known as the generalized Bargmann-Fock space) with a positive measure symbol $\nu$ and $0<s\leq 1$.