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Zbl 1235.47021
The Gelfand-Phillips property in closed subspaces of some operator spaces.
(English)
[J] Banach J. Math. Anal. 5, No. 2, 84-92, electronic only (2011). ISSN 1735-8787/e

A subset $A$ of a Banach space $X$ is called limited if every w*-null sequence in $X^*$ converges uniformly on $A$. The authors introduce limited completely continuous (lcc) operators as follows: a linear operator $T: X \to Y$ is lcc if it maps limited weakly null sequences into norm null ones. In this terminology, a Banach space has the Gelfand-Phillips property if the identity operator in this space is lcc. \par The authors study properties of lcc operators and their relations with other classes of operators. Some conditions for the Gelfand-Phillips property of subspaces of some operator spaces are given in terms of limited complete continuity of evaluation operators.
MSC 2000:
*47B07 Operators defined by compactness properties
46B10 Duality and reflexivity in normed spaces
47L20 Operator ideals
46B28 Normed linear spaces of linear operators, etc.

Keywords: Gelfand-Phillips property; Schur property; limited set; evaluation operator; operator ideal

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