New criteria for the weak Radon-Nikodým property related to set-valued operators
type: Item
instance of: scholarly article
title: New criteria for the weak Radon-Nikodým property related to set-valued operators
zbMATH Open document ID: 1352.46018
DOI: 10.1216/RMJ-2015-45-5-1511
author: Keun Young Lee
published in: Rocky Mountain Journal of Mathematics
publication date: 2016-02-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1453817252
review text: 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 …
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 6538957
zbMATH Keywords: weak Radon-Nikodym property
zbMATH Keywords: weakly fragmented set-valued operator
zbMATH Keywords: weakly single-valued operator
reviewed by: Minos Petrakis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6538957