On asymptotic and essential Toeplitz and Hankel integral operator

TL;DR

This paper characterizes the asymptotic behavior and essential properties of Toeplitz and Hankel integral operators on Hilbert space, providing complete descriptions of symbols for which these operators are essentially of these types.

Contribution

It offers a comprehensive characterization of when generalized integral operators are asymptotic and essentially Toeplitz or Hankel, advancing understanding of their structure on Hilbert spaces.

Findings

Characterization of uniform, strong, and weak asymptotic Toeplitz and Hankel operators.

Complete description of symbols for essentially Toeplitz and Hankel operators.

Conditions under which these operators exhibit specific asymptotic behaviors.

Abstract

In this article we consider the generalized integral operators acting on the Hilbert space $H^2$. We characterize when these operators are uniform, strong and weakly asymptotic Toeplitz and Hankel operators. Moreover we completely describe the symbols $g$ for which these operators are essentially Hankel and essentially Toeplitz.