TY - GEN

T1 - 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)

AU - Caponetti, Diana

PY - 2010

Y1 - 2010

N2 - 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’sdouble limit criterion. They also deal with the quantitative characteristicsk(H) andck(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 sequencesin H. The authors prove the following chain of inequalitiesck(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, eachcollection of closed convex subsets of the space with empty intersection has a countable subcollectionwith 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 ofadjoint operators in Banach spaces and for the classical Grothendieck’s characterization of weakcompactness in spaces C(K).

AB - 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’sdouble limit criterion. They also deal with the quantitative characteristicsk(H) andck(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 sequencesin H. The authors prove the following chain of inequalitiesck(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, eachcollection of closed convex subsets of the space with empty intersection has a countable subcollectionwith 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 ofadjoint operators in Banach spaces and for the classical Grothendieck’s characterization of weakcompactness in spaces C(K).

KW - Measure of noncompactness

KW - Measure of noncompactness

UR - http://hdl.handle.net/10447/55820

M3 - Other contribution

ER -