Interpolation of Operator Spaces

Abstract

We develop the real interpolation theory for operator spaces. We show that the main theorems in the classical real interpolation theory (for Banach spaces) hold in our operator space setting. These include the equivalence, duality and reiteration theorems. As an application of the duality theorem, we prove that the real interpolation operator space of parameters (1/2, 2) between an operator space and its antidual is completely isomorphic to an operator Hilbert spaceOH(I), discovered recently by G. Pisier. We also present an outline of a general interpolation theory in the category of operator spaces. In particular, we construct the operator space version of the orbit and coorbit functors, and extend to operator spaces the classical Aronzsajn–Gagliardo theorem. Our development complements some recent works of Pisier.