A note on relative compactness in \(K(X,Y)\) (Q306968)

Let \(X\) and \(Y\) be Banach spaces. The space of all compact operators between \(X\) and \(Y\) is denoted by \(K(X, Y)\). In this paper, the author reobtains a result of \textit{T. W. Palmer} [Proc. Am. Math. Soc. 20, 101--106 (1969; Zbl 0165.47603)] for relatively compact subsets of \(K(X, Y)\) by using a characterization of weakly precompact subsets of \(K(X,Y)\) due to the author [Commentat. Math. Univ. Carol. 56, No. 3, 319--329 (2015; Zbl 1349.46019)]. Furthermore, some necessary and sufficient conditions for the Dunford-Pettis relatively compact property of the spaces \(K(X, Y)\) and \(K(X, Y^*)\) are given.