Toeplitz operators defined by sesquilinear forms: Bergman space case

TL;DR

This paper extends the definition of Toeplitz operators in the Bergman space to include highly singular symbols like measures, distributions, and hyper-functions, broadening their applicability beyond traditional limits.

Contribution

It introduces a generalized framework for Toeplitz operators in the Bergman space that encompasses singular symbols previously not covered.

Findings

Extended Toeplitz operator definition includes measures, distributions, hyper-functions

Broadened applicability in Bergman space for highly singular symbols

Framework covers cases where traditional definitions fail

Abstract

The definition of Toeplitz operators in the Bergman space $A^2(D)$ of square integrable analytic functions in the unit disk in the complex plane is extended in such way that it covers many cases where the traditional definition does not work. This includes, in particular, highly singular symbols such as measures, distributions, and certain hyper-functions.