Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre

Abstract

Let $X=(X_0,X_1)$ and $Y=(Y_0,Y_1)$ be Banach couples and suppose $T:X\to Y$ is a linear operator such that $T:X_0\to Y_0$ is compact. We consider the question whether the operator $T:[X_0,X_1{]}_{\theta }\to [Y_0,Y_1{]}_{\theta }$ is compact and show a positive answer under a variety of conditions. For example it suffices that $X_0$ be a UMD-space or that $X_0$ is reflexive and there is a Banach space so that $X_0=[W,X_1{]}_{\alpha }$ for some $0<\alpha <1.$