Complex Interpolation between Hilbert, Banach and Operator spaces

TL;DR

This paper investigates the structure of certain Banach spaces related to complex interpolation, operator norms, and Hilbertian spaces, providing characterizations and descriptions of these spaces and their properties.

Contribution

It characterizes Banach spaces with specific operator norm decay properties and describes complex interpolation spaces involving operator and measure spaces, extending previous results.

Findings

Characterization of spaces with operator norm decay as quotients of ultraproducts of $ heta$-Hilbertian spaces

Description of complex interpolation spaces for regular operators and measure spaces

Extensions to Schur multipliers 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 $\vp\to \Delta_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T\colon L_2\to L_2$ 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 $\le \Delta_X(\vp)$ on $L_2(X)$. We show that $\Delta_X(\vp)\in O(\vp^\alpha)$ 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 \[ (B_{r}(L_2(\mu), B(L_2(\mu))^\theta. \] We show that $T\colon L_2(\mu)\to L_2(\mu)$ belongs to this space iff $T\otimes id_X$ is bounded on $L_2(X)$ for any $\theta$-Hilbertian space $X$. More generally, we are able to describe the spaces $$ (B(\ell_{p_0}), B(\ell_{p_1}))^\theta {\rm or} (B(L_{p_0}), B(L_{p_1}))^\theta $$ 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 $\|\mu\|_{M(G)} = |\mu|(G)$ (resp. \[ \|\mu\|_{PM(G)} = \sup\{|\hat\mu(\gamma)| \big| \gamma\in\hat G\}). \] 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.