On compactness theorems for logarithmic interpolation methods

Abstract

Let (A0;A1) be a Banach couple, (B0;B1) a quasi-Banach couple, 0 < q <= ∞ and T a linear operator. We prove that if T : A0 -> B0 is bounded and T : A1 -> B1 is compact, then the interpolated operator by the logarithmic method T : (A0,A1)1;q;A -> (B0;B1)1;q;A is compact too. This result allows the extension of some limit variants of Krasnosel'skii's compact interpolation theorem.