| Summary: | The Gelfand-Phillips property in closed subspaces of some operator spaces. (English summary) Banach J. Math. Anal. 5 (2011), no. 2, 84–92. The authors investigate the operator ideal on Banach spaces consisting of the limitedly completely continuous operators, and they discuss the Gelfand-Phillips property of certain closed subspaces of spaces of bounded operators. Recall that the subset A ⊂ X is limited if limn sup a∈A |〈a, x ∗ n〉 | = 0 for any weak ∗-null sequence (x ∗ n) ⊂ X ∗. The Banach space X has the Gelfand-Phillips property if any limited set A ⊂ X is relatively compact. Let X and Y be Banach spaces, and let L(X, Y) be the space of bounded operators X → Y. The authors call an operator T ∈ L(X, Y) limitedly completely continuous (abbreviated lcc) if limn ‖T xn ‖ = 0 for all limited, weak-null sequences (xn) ⊂ X. Sample results: (1) T is lcc if and only if T maps limited sets A ⊂ X to relatively compact sets T (A). (2) Assume that M ⊂ L(X, Y) is a closed subspace consisting of limited operators (that is, operators T for which T BX is limited, where BX is the closed unit ball of X), and that the closed linear span [T x: T ∈ M, x ∈ BX] has the Gelfand-Phillips property. Then M has the Gelfand-Phillips property whenever the operators ψy ∗: M → X ∗ are lcc for all y ∗ ∈ Y ∗ , where ψy ∗(T) = T ∗ y ∗ for T ∈ M and y ∗ ∈ Y ∗.
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