A Fuglede type theorem for Fourier multiplier operators

TL;DR

This paper proves a Fuglede type theorem for Fourier multiplier operators on translation invariant Banach function spaces over compact abelian groups, extending classical results to a broader functional analytic setting.

Contribution

It introduces a Fuglede type theorem specifically for Fourier multiplier operators on translation invariant Banach function spaces with order continuous norm.

Findings

Establishes a Fuglede type theorem in this new setting

Extends classical spectral theorems to Banach function spaces

Provides a framework for further spectral analysis of Fourier multipliers

Abstract

In this note a Fuglede type theorem is proved for Fourier multiplier operators on translation invariant Banach function spaces with order continuous norm over compact abelian groups.