(s)-nuclear sets and operators (Q1077689)

A set D in a Banach space E (with the unit ball \(B_ E)\) is called (s)- nuclear if its sequence of Kolmogorov diameters \(\delta_ n(D)=\inf \{r>0:\) \(D\subset F_ n+rB_ E\), \(F_ n\subset E\) a n-dimensional subspace\(\}\) satisfies \((\delta_ n(D))_ 1^{\infty}\in (s)\), where (s) is the space of rapidly decreasing sequences. An operator \(T\in {\mathcal L}(E,F)\) is (s)-nuclear if \(TB_ E\) is (s)-nuclear; similarly T is compact if \(TB_ E\) is (relatively) compact. The paper demonstrates considerable similarities in the behaviour of compact and (s)-nuclear operators: many previously known properties of compact operators, such as characterization of compact sets of compact operators and of collective compactness, are shown to have (s)-nuclear counterparts. However, in the (s)-nuclear case the Ascoli-Arzela theorem does not hold and hence other methods such as finitely dimensional projections must be used.