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 if 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).
| ISBN: | 9780821848425 |
| Publication date: | 30th November 2010 |
| Author: | Gilles Pisier |
| Publisher: | American Mathematical Society |
| Format: | Paperback |
| Pagination: | 78 pages |
| Series: | Memoirs of the American Mathematical Society |
| Genres: |
Calculus and mathematical analysis |
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 if 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).
Complex Interpolation Between Hilbert, Banach, and Operator Spaces features in the following genres: Calculus and mathematical analysis
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Complex Interpolation Between Hilbert, Banach, and Operator Spaces was written by Gilles Pisier and published by American Mathematical Society
Complex Interpolation Between Hilbert, Banach, and Operator Spaces has 78 pages
Yes it is part of Memoirs of the American Mathematical Society series