On products of some Toeplitz operators on polyanalytic Fock spaces

TL;DR

This paper investigates the boundedness of Toeplitz operator products on polyanalytic Fock spaces, providing necessary and sufficient conditions and characterizations via the Berezin transform.

Contribution

It establishes a complete characterization of bounded Toeplitz products on polyanalytic Fock spaces, extending Sarason's problem to this setting.

Findings

Derived necessary and sufficient conditions for boundedness

Characterized boundedness using the Berezin transform

Extended Sarason's problem to polyanalytic Fock spaces

Abstract

The purpose of this paper is to study the Sarason's problem on Fock spaces of polyanalytic functions. Namely, given two polyanalytic symbols $f$ and $g$, we establish a necessary and sufficient condition for the boundedness of some Toeplitz products $ T_{f}T_{\bar g}$ subjected to certain restriction on $f$ and $g$. We also characterize this property in terms of the Berezin transform.