Geometric characterization of $RN$-operators

O. I. Reinov

Matematičeskie zametki, Tome 22 (1977) no. 2, pp. 189-202.

Résumé

Let $X$ and $Y$ be Banach spaces and $T\in L(X,Y)$. An operator $T:X\to Y$ is called an $RN$-operator if it transforms every $X$-valued. measure $\overline m$ of bounded variation into a $Y$-valued measure having a derivative with respect to the variation of the measure $\overline m$. 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 $X_0$ of $X$ the set $(T|X_0)^*(Y^*)$ is separable (Theorem 2).