On semibounded Wiener-Hopf operators

TL;DR

This paper characterizes when semibounded Wiener-Hopf quadratic forms are closable in L^2, linking this property to the Fourier transform of absolutely continuous measures, and introduces minimal assumptions for defining related operators.

Contribution

It provides a necessary and sufficient condition for the closability of semibounded Wiener-Hopf quadratic forms based on the measure's absolute continuity, extending the understanding of these operators.

Findings

Closable Wiener-Hopf quadratic forms correspond to Fourier transforms of absolutely continuous measures.

The paper introduces a continuous analogue of the Riesz Brothers theorem.

Minimal assumptions are identified for defining semibounded Wiener-Hopf operators.

Abstract

We show that a semibounded Wiener-Hopf quadratic form is closable in the space $L^2({\Bbb R}_{+})$ if and only if its integral kernel is the Fourier transform of an absolutely continuous measure. This allows us to define semibounded Wiener-Hopf operators and their symbols under minimal assumptions on their integral kernels. Our proof relies on a continuous analogue of the Riesz Brothers theorem obtained in the paper.