Complex Interpolation between Hilbert, Banach and Operator Spaces Academic Article uri icon


  • 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.

published proceedings


author list (cited authors)

  • Pisier, G.

citation count

  • 22

complete list of authors

  • Pisier, Gilles

publication date

  • November 2010