# Toeplitz operators with singular symbols in polyanalytic Bergman spaces on the half-plane

## TL;DR

This paper studies Toeplitz operators with highly singular symbols in polyanalytic Bergman spaces on the half-plane, establishing conditions for their boundedness and compactness, and reducing their analysis to simpler analytic cases.

## Contribution

It introduces a sesquilinear form approach for Toeplitz operators with singular symbols and develops a reduction method to analyze polyanalytic cases via analytic Bergman spaces.

## Key findings

- Conditions for boundedness of Toeplitz operators with singular symbols
- Criteria for compactness of these operators
- Reduction technique linking polyanalytic and analytic Bergman spaces

## Abstract

Using the approach based on sesquilinear forms, we introduce Toeplitz operator in the analytic Bergman space on the upper half-plane with strongly singular symbols, derivatives of measures. Conditions for boundedness and compactness of such operators are found. A procedure of reduction of Toeplitz operators in Bergman spaces of polyanalytic functions to operators with singular symbols in the analytic Bergman space by means of the creation-annihilation structure is elaborated, which leads to the description of the properties of the former operators