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Some classes of sets in Banach spaces and the topological characterization of operators of type RN. (Russian) Zbl 0406.46013
Summary: The purpose of this note is to discuss properties of bounded subsets of Banach spaces related to the Grothendieck’s notion of equimeasurability of sets of functions. A corollary of our main results is that given any Banach space \(X\) and \(Y\) and an operator \(T\in L(X,Y)\), the mapping \(T\) is an operator of type RN if and only if for each \(p\geq 0\)b and every \(p\)-absolute summing transformation \(U\) from any Banach space \(Z\) into \(X\) the composition \(TU\) is an approximatively \(p\)-radonifying transformation from \(Z\) into \(Y\).

MSC:
46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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