Operators in the Fock-Toeplitz algebra

TL;DR

This paper explores various bounded operators on the Fock space, examining their inclusion in the Toeplitz algebra through classical and new methods, with insights from quantum harmonic analysis.

Contribution

It provides new theorems and links between known results on operator classes in the Fock space, integrating classical and recent approaches with quantum harmonic analysis.

Findings

Characterization of when operators belong to the Toeplitz algebra

New theorems linking classical and recent operator results

Application of quantum harmonic analysis to operator theory

Abstract

We consider various classes of bounded operators on the Fock space $F^2$ of Gaussian square integrable entire functions over the complex plane. These include Toeplitz (type) operators, weighted composition operators, singular integral operators, Volterra-type operators and Hausdorff operators and range from classical objects in harmonic analysis to more recently introduced classes. As a leading problem and closely linked to well-known compactness characterizations we pursue the question of when these operators are contained in the Toeplitz algebra. This paper combines a (certainly in-complete) survey of the classical and more recent literature including new ideas for proofs from the perspective of quantum harmonic analysis (QHA). Moreover, we have added a number of new theorems and links between known results.