Complex Interpolation between Hilbert, Banach and Operator spaces

Abstract

Motivated by a question of Vincent Lafforgue, we study the Banach spaces $X$ satisfying the following property: there is a function $\vp\to Δ_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T:L_2\to L_2$ with $\|T\|\le \vp$ that is simultaneously contractive (i.e. of norm $\le 1$) on $L_1$ and on $L_{\infty }$ must be of norm $\le Δ_X(\vp)$ on $L_2(X)$. We show that $Δ_X(\vp)\in O(\vp^α)$ for some $\alpha >0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $\theta$-Hilbertian spaces for some $\theta >0$ (see Corollary \ref{comcor4.3}), where $\theta$-Hilbertian is meant in a slightly more general sense than in our previous paper \cite{P1}. Let $B_r(L_2(\mu ))$ be the space of all regular operators on $L_2(\mu )$. We are able to describe the complex interpolation space \[ (B_{r}(L_2(μ), B(L_2(μ))^θ. \] We show that $T:L_2(\mu )\to L_2(\mu )$ belongs to this space iff $T\otimes i{d}_X$ is bounded on $L_2(X)$ for any $\theta$-Hilbertian space $X$. More generally, we are able to describe the spaces $$(B({\ell }_{p_0}),B({\ell }_{p_1}){)}^{\theta }\mathrm{or}(B(L_{p_0}),B(L_{p_1}){)}^{\theta }$$ for any pair $1\le p_0,p_1\le \infty$ and $0<\theta <1$. In the same vein, given a locally compact Abelian group $G$, let $M(G)$ (resp. $PM(G)$) be the space of complex measures (resp. pseudo-measures) on $G$ equipped with the usual norm $\Vert \mu {\Vert }_{M(G)}=|\mu |(G)$ (resp. \[ \|μ\|_{PM(G)} = \sup\{|\hatμ(γ)| \big| γ\in\hat G\}). \] We describe similarly the interpolation space $(M(G),PM(G){)}^{\theta }$. Various extensions and variants of this result will be given, e.g. to Schur multipliers on $B({\ell }_2)$ and to operator spaces.

arXiv commentA detailed proof of Hernandez Theorem 4.6 has been added, as well as various minor improvements and clarifications