Toeplitz versus Hankel: semibounded operators

TL;DR

This paper compares Toeplitz and Hankel operators, establishing conditions for their quadratic forms to be closable and enabling their definition as self-adjoint operators with minimal assumptions.

Contribution

It provides necessary and sufficient conditions for the closability of quadratic forms of semibounded Toeplitz and Hankel operators, advancing their mathematical understanding.

Findings

Criteria for quadratic form closability established

Domains of closed quadratic forms characterized

Operators defined as self-adjoint under minimal assumptions

Abstract

Our goal is to compare various results for Toeplitz $T$ and Hankel $H$ operators. We consider semibounded operators and find necessary and sufficient conditions for their quadratic forms to be closable. This property allows one to define $T$ and $H$ as self-adjoint operators under minimal assumptions on their matrix elements. We also describe domains of the closed Toeplitz and Hankel quadratic forms.