A Fuglede type theorem for Fourier multiplier operators

More Info


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.