Best possible compactness results of Lions-Peetre type

If T : A(0) --> B boundedly and T : A(1) --> B compactly, then a result of Lions-Peetre shows that T : A --> B compactly for a certain class of spaces A which are intermediate with respect to A(0) and A(1). We investigate to what extent such results can hold for arbitrary intermediate spaces A. The 'dual' case of an operator S such that S : X --> Y-0 boundedly and S : X --> Y-1 compactly, is also considered, as well as similar questions for other closed operator ideals.