MR2502017 (2010c:46055) Angosto, C.; Cascales, B. Measures of weak noncompactness in Banach spaces. Topology Appl. 156 (2009), no. 7, 1412--1421. (Reviewer: Diana Caponetti) 46B99 (46A50 47B07 47H09 54C35)

Abstract

The authors consider for a bounded subset H of a Banach space E the De Blasi measure of weak noncompactness w(H) and the measure of weak noncompactness g(H) based on Grothendieck’s double limit criterion. They also deal with the quantitative characteristics k(H) and ck(H) which represent, respectively, the worst distance to E of the weak*-closure of H in the bidual of E and the worst distance to E of the sets of weak*-cluster points in the bidual of E of sequences in H. The authors prove the following chain of inequalities ck(H) < = k(H) < = g(H) < = 2ck(H) < = 2k(H) < = 2w(H), which, in particular, shows that ck, k and g are equivalent. The authors show that ck = k in the class of Banach spaces with Corson property C (i.e, each collection of closed convex subsets of the space with empty intersection has a countable subcollection with empty intersection), but they also give an example for which k(H) = 2ck(H). Moreover, they obtain quantitative counterparts for of Gantmacher’s theorem about weak compactness of adjoint operators in Banach spaces and for the classical Grothendieck’s characterization of weak compactness in spaces C(K).