# Polyanalytic Toeplitz operators: isomorphisms, symbolic calculus and approximation of Weyl operators

## TL;DR

This paper extends Toeplitz quantization to polyanalytic functions, establishing isomorphisms between function spaces, deriving an asymptotic symbolic calculus, and approximating Weyl operators using polyanalytic Toeplitz operators.

## Contribution

It introduces a generalized framework for polyanalytic Toeplitz operators, extending known analytic results to a broader polyanalytic setting and providing new asymptotic expansion techniques.

## Key findings

- Established isomorphism theorem for polyanalytic Toeplitz operators.
- Derived an asymptotic symbol calculus for these operators.
- Presented an asymptotic expansion of Weyl operators in terms of polyanalytic Toeplitz operators.

## Abstract

We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.