A Fuglede type theorem for Fourier multiplier operators

Ben de Pagter; Werner J.  Ricker

doi:10.1016/j.indag.2022.10.005

A Fuglede type theorem for Fourier multiplier operators

Ben de Pagter, Werner J. Ricker

Analysis

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Abstract

Let E be a translation invariant Banach function space over an infinite compact abelian group G and M_φ be a Fourier multiplier operator (with symbol φ) acting on E. It is assumed that E has order continuous norm and that E is reflection invariant (which ensures that φ̄ is also a multiplier symbol for E). The following Fuglede type theorem is established. Whenever T is a bounded linear operator on E satisfying M_φT=TM_φ, then also M_φ̄T=TM_φ̄.

Original language	English
Pages (from-to)	259-273
Number of pages	15
Journal	Indagationes Mathematicae
Volume	34 (2023)
Issue number	2
DOIs	https://doi.org/10.1016/j.indag.2022.10.005
Publication status	Published - 2022

Keywords

Compact abelian group
Fourier multiplier operator
Reflection invariance
Translation invariant Banach function space

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10.1016/j.indag.2022.10.005

1-s2.0-S0019357722000854-mainFinal published version, 342 KBLicence: CC BY

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AB - Let E be a translation invariant Banach function space over an infinite compact abelian group G and Mφ be a Fourier multiplier operator (with symbol φ) acting on E. It is assumed that E has order continuous norm and that E is reflection invariant (which ensures that φ̄ is also a multiplier symbol for E). The following Fuglede type theorem is established. Whenever T is a bounded linear operator on E satisfying MφT=TMφ, then also Mφ̄T=TMφ̄.

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KW - Fourier multiplier operator

KW - Reflection invariance

KW - Translation invariant Banach function space

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A Fuglede type theorem for Fourier multiplier operators

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