Unconditionality, Fourier multipliers and Schur multipliers

TL;DR

This paper investigates the relationship between Fourier multipliers, Banach space structures, and operator space properties on locally compact abelian groups, revealing conditions for Hilbert space isomorphism and boundedness distinctions.

Contribution

It establishes that certain bounded Fourier multiplier properties characterize Hilbert spaces and identifies the existence of non-completely bounded multipliers on L^p spaces for p ≠ 2, also analyzing unconditionality via Schur multipliers.

Findings

Banach space X is Hilbert if all Fourier multipliers extend to bounded operators on X.

Existence of bounded Fourier multipliers on L^p(G) that are not completely bounded for p ≠ 2.

Characterization of operator spaces that are completely isomorphic to operator Hilbert spaces.

Abstract

Let $G$ be an infinite locally compact abelian group. If $X$ is Banach space, we show that if every bounded Fourier multiplier $T$ on $L^2(G)$ has the property that $T\ot Id_X$ is bounded on $L^2(G,X)$ then the Banach space $X$ is isomorphic to a Hilbert space. Moreover, if $1<p<\infty$, $p\not=2$, we prove that there exists a bounded Fourier multiplier on $L^p(G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions to determine if an operator space is completely isomorphic to an operator Hilbert space.