Complex Interpolation between Hilbert, Banach and Operator Spaces
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Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function X() tending to zero with > 0 such that every operator T: L 2 L 2 with ||T|| that is simultaneously contractive (i.e. of norm 1) on L1 and on L must be of norm X() on L 2(X). We show that X() O() for some > 0 iff X is isomorphic to a quotient of a subspace of an ultraproduct of -Hilbertian spaces for some > 0 (see Corollary 6.7), where -Hilbertian is meant in a slightly more general sense than in our previous paper (1979). Let B r(L2()) be the space of all regular operators on L 2 (). We are able to describe the complex interpolation space (B r(L 2()), B(L 2())). We show that T: L 2() L2() belongs to this space iff T id X is bounded on L 2(X) for any -Hilbertian space X. More generally, we are able to describe the spaces (B(l po), B(l pl)) or (B(L po), B(L pl)) for any pair 1 p 0,p 1 00 and 0 < < 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 ||||m(g) = ||(G) (resp. |||| PM(G) = sup{ | () | I }). We describe similarly the interpolation space (M(G), PM(G)) . Various extensions and variants of this result will be given, e.g. to Schur multipliers on B(l 2) and to operator spaces. 2010 American Mathematical Society.