On compact sets of compact operators on Banach spaces not containing a copy of \(l^1\). (English) Zbl 1278.46019

The author gives a new proof of a result of F. Mayoral [Proc. Am. Math. Soc. 129, No. 1, 79–82 (2001; Zbl 0961.47010)] that gives a necessary condition for a subset \(M\) of the space of compact operators \({\mathcal K}(X,Y)\) to be compact, when the Banach space \(X\) does not contain an isomorphic copy of \(\ell^1\). The key ingredient is a version of the ArzelĂ -Ascoli theorem due to G. Nagy [Real Anal. Exch. 32(2006–2007), No. 2, 583–585 (2007; Zbl 1139.46026)].


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