Toeplitz and related operators on polyanalytic Fock spaces

TL;DR

This paper characterizes compact and Fredholm operators on polyanalytic Fock spaces using limit operators, generalizes the Bauer-Isralowitz theorem with a matrix-valued transform, and provides criteria for Toeplitz and Hankel operator compactness.

Contribution

It introduces a new characterization of operators on polyanalytic Fock spaces and extends the Bauer-Isralowitz theorem with a matrix-valued Berezin transform.

Findings

Compactness of Toeplitz and Hankel operators is independent of polyanalytic order.

The generalized Bauer-Isralowitz theorem applies to matrix-valued transforms.

Necessary and sufficient conditions for operator compactness are established.

Abstract

We give a characterization of compact and Fredholm operators on polyanalytic Fock spaces in terms of limit operators. As an application we obtain a generalization of the Bauer-Isralowitz theorem using a matrix valued Berezin type transform. We then apply this theorem to Toeplitz and Hankel operators to obtain necessary and sufficient conditions for compactness. As it turns out, whether or not a Toeplitz or Hankel operator is compact does not depend on the polyanalytic order. For Hankel operators this even holds on the true polyanalytic Fock spaces.