On semibounded Toeplitz operators

TL;DR

This paper characterizes when semibounded Toeplitz quadratic forms are closable in ll^2 space, linking this property to Fourier coefficients of absolutely continuous measures, and describes their domains.

Contribution

It provides a necessary and sufficient condition for the closability of semibounded Toeplitz quadratic forms based on measure theory.

Findings

Closable forms correspond to Fourier coefficients of absolutely continuous measures

The domain of the closed form is explicitly described

Enables defining semibounded Toeplitz operators with minimal assumptions

Abstract

We show that a semibounded Toeplitz quadratic form is closable in the space $\ell^2({\Bbb Z}_{+})$ if and only if its matrix elemens are Fourier coefficients of an absolutely continuous measure. We also describe the domain of the corresponding closed form. This allows us to define semibounded Toeplitz operators under minimal assumptions on their matrix elements.