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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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