Complex Interpolation Between Hilbert Banach And Operator Spaces
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Complex Interpolation between Hilbert, Banach and Operator Spaces
Author: Gilles Pisier
language: en
Publisher: American Mathematical Soc.
Release Date: 2010-10-07
Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces $X$ satisfying the following property: there is a function $\varepsilon\to \Delta_X(\varepsilon)$ tending to zero with $\varepsilon>0$ such that every operator $T\colon \ L_2\to L_2$ with $\T\\le \varepsilon$ that is simultaneously contractive (i.e., of norm $\le 1$) on $L_1$ and on $L_\infty$ must be of norm $\le \Delta_X(\varepsilon)$ on $L_2(X)$. The author shows that $\Delta_X(\varepsilon) \in O(\varepsilon^\alpha)$ 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 6.7), where $\theta$-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).
Martingales in Banach Spaces
Author: Gilles Pisier
language: en
Publisher: Cambridge University Press
Release Date: 2016-06-06
This book focuses on applications of martingales to the geometry of Banach spaces, and is accessible to graduate students.
The Operator Hilbert Space $OH$, Complex Interpolation and Tensor Norms
Author: Gilles Pisier
language: en
Publisher: American Mathematical Soc.
Release Date: 1996
In the recently developed duality theory of operator spaces, bounded operators are replaced by 'completely bounded' ones, isomorphism by 'complete isomorphisms' and Banach spaces by 'operator spaces'. This allows for distinguishing between the various ways in which a given Banach space can be embedded isometrically into [italic capital]B([italic capital]H) (with H being Hilbert). One of the main results is the observation that there is a central object in this class: there is a unique self dual Hilbertian operator space (which we denote by [italic capitals]OH) which seems to play the same central role in the category of operator spaces that Hilbert spaces play in the category of Banach spaces.