Unconditionality, Fourier multipliers and Schur multipliers

Cédric Arhancet¹

¹ Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon Cedex, France

Colloquium Mathematicum, Tome 127 (2012) no. 1, pp. 17-37.

Résumé

Let $G$ be an infinite locally compact abelian group and $X$ be a Banach space. We show that if every bounded Fourier multiplier $T$ on $L^2(G)$ has the property that $T\otimes {\rm Id}_X$ is bounded on $L^2(G,X)$ then $X$ is isomorphic to a Hilbert space. Moreover, we prove that if $1 p \infty $, $p\not =2$, then 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 for an operator space to be completely isomorphic to an operator Hilbert space.

Détail

DOI : 10.4064/cm127-1-2

Keywords: infinite locally compact abelian group banach space every bounded fourier multiplier has property otimes bounded isomorphic hilbert space moreover prove infty there exists bounded fourier multiplier which completely bounded finally examine unconditionality point view schur multipliers precisely several necessary sufficient conditions operator space completely isomorphic operator hilbert space