On compactness in complex interpolation (Q455459)

Let \((X_0,X_1)\) and \((Y_0,Y_1)\) be Banach couples and let \(T\) be a linear operator such that \(T:X_0\to Y_0\) compactly and \(T:X_1\to Y_1\) boundedly. It is a major problem in the theory of interpolation of operators to decide whether \(T:X_\vartheta\to Y_\vartheta\) compactly, where \(X_\vartheta =[X_0,X_1]_\vartheta\) and \(Y_\vartheta =[Y_0,Y_1]_\vartheta\) denote the complex interpolation spaces. For many important cases, an affirmative answer is known; however, whether this is true in general remains open (note that in a recent paper, the general case has been claimed to have an affirmative answer; however, its proof seems to be incorrect, see [\textit{N. C. Freire}, Vietnam J. Math. 41, No. 1, 115--120 (2013; Zbl 1269.46013)]. In the particular cases where \(X_0=X_1\) or \(Y_0=Y_1\), it has been shown that the answer is positive, without further requirements on the spaces. The article under review deals with similar results for operators \(T(z)\) acting on the strip \(S=\{z \in \mathbb{C}:0 \leq \text{Re\,} z \leq 1 \}\) with values in \(\mathcal{L}(X_\Delta,Y_\Sigma)\). It turns out that in the case \(Y_0=Y_1\), compactness on a subset of the boundary of positive measure yields compactness within the interior of the strip. The author also provides a counterpart for the case \(X_0=X_1\), albeit requiring more restrictive hypotheses.