Abstract
Let X and Y be Banach spaces andt∃l (x, y). An operator T: X → Y is called an RN-operator if it transforms every X-valued. measure ¯m of bounded variation into a Y-valued 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 1). It is also proved that the adjoint operator is an RN-operator if and only if for every separable subspace Xo of X the set (T|Xo)*(Y*) is separable (Theorem 2).
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Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 189–202, August, 1977.
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Reinov, O.I. Geometric characterization of RN-operators. Mathematical Notes of the Academy of Sciences of the USSR 22, 597–604 (1977). https://doi.org/10.1007/BF01780967
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DOI: https://doi.org/10.1007/BF01780967