On compactness theorems for logarithmic interpolation methods

Let $(A_{0}, A_{1})$ be a Banach couple, $(B_{0}, B_{1})$ a quasi-Banach couple, $0 \lt q\leq \infty $ and $T$ a linear operator. We prove that if $T: A_{0} \rightarrow B_{0}$ is bounded and $T: A_{1} \rightarrow B_{1}$ is compact, then the interpolated operator by the logarithmic method $T: (A_{0}, A_{1})_{1,q,\mathbb{A}} \rightarrow (B_{0}, B_{1})_{1,q, \mathbb{A}}$ is compact too. This result allows the extension of some limit variants of Krasnosel’skiǐ’s compact interpolation theorem.