Memoirs of the American Mathematical Society 2010; 78 pp; softcover Volume: 208 ISBN10: 0821848429 ISBN13: 9780821848425 List Price: US$68 Individual Members: US$40.80 Institutional Members: US$54.40 Order Code: MEMO/208/978
 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). Table of Contents  Introduction
 Preliminaries. Regular operators
 Regular and fully contractive operators
 Remarks on expanding graphs
 A duality operators/classes of Banach spaces
 Complex interpolation of families of Banach spaces
 \(\pmb{\theta}\)Hilbertian spaces
 Arcwise versus not arcwise
 Fourier and Schur multipliers
 A characterization of uniformly curved spaces
 Extension property of regular operators
 Generalizations
 Operator space case
 Generalizations (Operator space case)
 Examples with the Haagerup tensor product
 References
