New criteria for the weak Radon-Nikodým property related to set-valued operators
A subset \(K\) of a Banach space has the weak Radon-Nikodým property (wRNP) if, for every finite complete measure space \((\Omega,\Sigma,\mu)\) and each \(\mu\)-continuous \(X\)-valued countably additive vector measure \(\nu:\Sigma\to X\) of bounded variation such that \(\nu(E)\in\mu(E)\cdot K\) for all \(E\) in \(\Sigma\), there exists a \(K\)-valued Pettis integrable function \(f:\Omega\to K\) such that \(\nu(E)=\mathrm{Pettis}-\int f\, d\mu\) for all \(E\in\Sigma\). In the case where \(K\) is a weak* compact subset of the dual of the Banach space \(X\), \(K\) has the wRNP iff, for every subset \(A\) of \(K\) and every \(\varepsilon>0\), there is a non-empty \(w^\ast\)-slice \(S\) of \(A\) (i.e., there are \(x\in X\) and \(c>0\) such that \(S=\{x^\ast A:x^\ast(x)>c\})\) so that \(\operatorname{diam}(S)<\varepsilon\). It is well known that the unit ball of the dual of a Banach space \(X\) has the wRNP if and only if \(X\) contains no isomorphic copy of \(l_1\). The author looks at the wRNP property of \(w^\ast\)-compact subsets of a dual Banach space \(X^\ast\) from the point of view of multivalued functions. The following theorem is shown: If the unit ball of \(X^\ast\) has the wRNP but fails the RNP, then there exists a set-valued operator \(T:X\to 2^{(X^\ast)}\) which is weakly fragmented but not fragmented in norm. The main Theorem 4.7 contains four conditions equivalent to the wRNP for an \(w^\ast\)-compact subset \(K\) of a dual space. We state here only a small portion of the statements of Theorem 4.7. Let \(X,Y\) be Banach spaces and \(K\) be an \(w^\ast\)-compact subset of \(X^\ast\). Then \(K\) has the wRNP if and only if, for every \(T:Y\to 2^{(X)^\ast}\) upper semicontinuous with values in \(w^\ast\)-compact subsets of \(X^\ast\), with minimal graph, such that the set \(\{y\in Y:T(y)\cap K\neq\emptyset\}\) is dense in \(Y\) and for every \(x^{\ast\ast}\in X^{\ast\ast}\), there is a dense \(G_\delta\) subset \(D\) of \(Y\) such that \(x^{\ast\ast}T\) is single-valued and upper semicontinuous at each point of \(D\). The paper also contains many nice propositions on exposed points, and on the relation of Gâteaux differentiability of convex functions and their subdifferentials.
scientific article; zbMATH DE number 6538957