A note on relative compactness in \(K(X,Y)\). (English) Zbl 1359.46017

Ghenciu, Ioana

Ann. Funct. Anal. 7, No. 3, 470-483 (2016).

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 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.

Reviewer: Jafar Zafarani (Isfahan)

MSC:

46B28	Spaces of operators; tensor products; approximation properties
46B50	Compactness in Banach (or normed) spaces

Keywords:

compact operators; Dunford-Pettis relative compact property; Gelfand-Phillips property; compact subset

Citations:

Zbl 0165.47603; Zbl 1349.46019