Weakly, sufficiently or strongly localized operators on the Fock space in \mathh C^n

TL;DR

This paper investigates four classes of localized operators on Fock space in complex n-dimensional space, establishing strict inclusions among them and providing examples of operators with specific localization properties.

Contribution

It introduces and compares four classes of localized operators on Fock space, proving strict inclusions and constructing examples to distinguish these classes.

Findings

First two classes are strictly contained in the next two.

Existence of a weakly localized operator not in the Toeplitz algebra.

Examples of composition operators that are sufficiently but not strongly localized.

Abstract

We study properties of the following four classes of operators on the Fock space in $\mathbb C^n:$ 1) weakly localized operators; 2) sufficiently localized operators in the sense of Xia and Zheng; 3) sufficiently localized operators; 4) strongly localized operators. In this respect, we examine composition operators, Toeplitz operators with a measure symbol whose total variation measure is a Fock-Carleson measure, and singular operators of convolution type introduced by Zhu, among others. We also provide a bounded operator which is not weakly localized and does not even belong to the Toeplitz algebra. Class 1) contains class 2), class 2) contains class 3), which clearly contains class 4). We prove that the first two inclusions are strict. Our proofs are in terms of singular operators of convolution type introduced by Zhu. The third inclusion was already known to be strict, as Wang, Cao and Zhu exhibited examples of composition operators which are sufficiently localized, but are not strongly localized. %As our main result, we show the existence of a singular operator of convolution type which is weakly localized, but is not sufficiently localized in the sense of Xia and Zheng. %The underlying question is whether the first two classes of operators coincide or not.