Interpolation of compact operators by general interpolation methods

Let a linear operator T act compactly from a Banach couple X→:=(X0,X1) in a Banach lattice couple B→:=(B0,B1). If both Bi have absolutely continuous norm, then any interpolation functor F preserves two-sided compactness of T. The same holds if only one of Bi satisfies the above condition and F is a quasipower regular functor. The similar results are true for operators acting from B→ in X→ under the above formulated assumptions for simultaneously B→ and its dual B→′.