The complex interpolation method preserves compactness of linear operators (Q1952327)

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. The author of the paper at hand claims to solve this problem in the positive. However, his proofs are incorrect or not conclusive. Theorem~3 deals with the special case \(Y_0=Y_1\). The mistake in the proof is the estimate of the diameter of \(\Omega_0\) (see the paper for the definition of this and other items), which is false; the author has mixed up the \(\exists\)- and the \(\forall\)-quantifier. Incidentally, if the estimate were correct, the author could have passed to the limit \(t\to0\) to obtain that this diameter is~\(0\). Theorem~4 deals with the special case \(X_0=X_1\). Here, there is only a minor glitch in the argument that can be remedied; the author has treated an infimum as if it were a minimum. As it happens, both these results are true after all; they can be found in Theorem~3.8.1 of the classical monograph [\textit{J.~Bergh} and \textit{J.~Löfström}, Interpolation spaces. An introduction. Berlin etc.: Springer-Verlag (1976; Zbl 0344.46071)]. Finally, Theorem~5 purports to solve the general case via a known reduction argument. There are various problems in the proof. For one thing, in the argument that both limiting operators are compact, the author seems to assume that \(T\) and \(R\) commute. Then, the argument that \(D\) is dense is false (but the statement is correct); it is based on the erroneous assumption that every \(L_1\)-function on the unit circle is even an \(L_2\)-function. Finally, there is no valid argument that the operator \(B\) actually maps \(W\) into \(Y_\theta\), which is the core of the whole proof. Hence the problem that this paper allegedly solves remains open. The reviewer cannot help wondering why none of the elementary mistakes pointed out above was detected during the refereeing process.