Interpolation of compact operators by general interpolation methods (Q2338949)

Let \(F\) be an interpolation functor and \(T\) be a linear operator acting continuously from a Banach couple \(\bar{X}=(X_{0},X_{1})\) in a Banach lattice couple \(\bar{B}=(B_{0},B_{1})\). Assume that \(T\) acts compactly from \(X_{i}\) in \(B_{i}\), \(i=0,1\). If each \(B_{i}\) has absolutely continuous norm, then \(T\) acts compactly between the interpolation spaces \(F(\bar{X})\) and \(F(\bar{B})\). The same holds if only one of the \(B_{i}\) satisfies the above condition and \(F\) is a quasipower regular functor. Similar results are true for operators acting from \(\bar{B}\) in \(\bar{X}\) under the above formulated assumptions simultaneously for \(\bar{B}\) and its dual \(\bar{B}^{\prime }\).