Best possible compactness results of Lions-Peetre type (Q2716986)

By a Banach couple \((A_0, A_1)\) is understood a pair of Banach spaces \(A_0\), \(A_1\) continuously embedded in some Hausdorff topological vector space. A Banach space \(A\) satisfying \(A_0\cap A\subseteq A\subseteq A_0+ A_1\) is called an intermediate space with respect to the couple \((A_0, A_1)\). Let \(T: A_0+ A_1\to B\) be a linear operator with values in a Banach space \(B\) and suppose that the restrictions of \(T\) to \(A_0\) and to \(A_1\) are respectively bounded and compact. The authors investigate conditions under which the restriction of \(T\) to any intermediate space \(A\) is compact.