Toeplitz operators in polyanalytic Bergman type spaces

TL;DR

This paper investigates Toeplitz operators in polyanalytic Bergman and Fock spaces, revealing how their properties can be understood through reduction to classical spaces with distributional symbols, highlighting differences from standard Bergman-Toeplitz operators.

Contribution

It introduces a reduction technique for Toeplitz operators in polyanalytic spaces to classical Bergman spaces with distributional symbols, expanding understanding of their properties.

Findings

Reduction of polyanalytic Toeplitz operators to classical spaces

Description of properties differing from standard Bergman-Toeplitz operators

Application to spaces on the disk, half-plane, and complex plane

Abstract

We consider Toeplitz operators in Bergman and Fock type spaces of polyanalytic $L^2\textup{-}$functions on the disk or on the half-plane with respect to the Lebesgue measure (resp., on $\mathbb{C}$ with the plane Gaussian measure). The structure involving creation and annihilation operators, similar to the classical one present for the Landau Hamiltonian, enables us to reduce Toeplitz operators in true polyanalytic spaces to the ones in the usual Bergman type spaces, however with distributional symbols. This reduction leads to describing a number of properties of the operators in the title, which may differ from the properties of the usual Bergman-Toeplitz operators.