Geometric characterization of RN-operators

Abstract

Let X and Y be Banach spaces and T E L(X. Y). An operator T: X § Y is called an RN-operator if it transforms every X-valued measure ~ of bounded variation into a Yvalued measure having a derivative with respect to the variation of the measure m. The notions of T-dentability and Ts-dentability of bounded sets in Banach spaces are introduced and in their terms are given conditions equivalent to the condition that T is an RN-operator (Theorem i).It is also proved that the adjoint operator is an RN-operator if and only if for every separable subspace X0 of X the set (TIX0)*(Y*) is separable (Theorem 2). RN-Operators, introduced in [i], are studied.Characteristics of RN-operators are deduced in certain geometrical terms (Theorem i), and the class of operators whose adjoint operators are RN-operators is described.In particular, we obtain a proof, which is not based on the Dunford--Phillips--Pettice theorem, of the fact -that every weakly compact operar tor is an RN-operator.Let X and Y be Banach spaces and L(X, Y) be the space of continuous linear operators from X into Y.If A is a bounded subset of X, then we will denote the closed convex hull of the set A by co(A) and the s-convex hull of the set A by s-co(A):

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

“…Here the map φ T (V ) is one-to-one, and we can follow the assertion from [8] for operators in Banach spaces: if U : X → Y is one-to-one then U is RN iff the Uimage of the unit ball is subset s-dentable). Another way is just to apply the main definition from this paper.…”

“…The geometrical characterization of RN-operators between Banach spaces has been given by following theorem in [8].…”

“…We present a theorem which says that, for a large class of l.c.v.s. E, F, if T : E → F is a linear continuous map, then the following are equivalent:Therefore, we have a generalization of Theorem 1 in [8], which gave a geometric characterization of RN-operators between Banach spaces.…”

“…Therefore, we have a generalization of Theorem 1 in [8], which gave a geometric characterization of RN-operators between Banach spaces.…”