Fine spectra of the finite Hilbert transform in function spaces

TL;DR

This paper analyzes the spectrum and fine spectra of the finite Hilbert transform on rearrangement invariant spaces, extending classical results from $L^p$ spaces to a broader class of function spaces.

Contribution

It provides a comprehensive description of the spectrum and fine spectra of the finite Hilbert transform on rearrangement invariant spaces with non-trivial Boyd indices, generalizing Widom's results.

Findings

Full spectrum description when Boyd indices coincide

Extension of spectral results from $L^p$ spaces to rearrangement invariant spaces

Characterization of the spectrum and fine spectra in these spaces

Abstract

We investigate the spectrum and fine spectra of the finite Hilbert transform acting on rearrangement invariant spaces over $(-1,1)$ with non-trivial Boyd indices, thereby extending Widom's results for $L^p$ spaces. In the case when these indices coincide, a full description of the spectrum and fine spectra is given.