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

Let X = ( X 0 , X 1 ) and Y = ( Y 0 , Y 1 ) be Banach couples and suppose T : X → Y is a linear operator such that T : X 0 → Y 0 is compact. We consider the question whether the operator T :[ X 0 , X 1 ] θ →[ Y 0 , Y 1 ] θ 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 ] α , for some 0<α<1.