Weak precompactness and property (V*) in spaces of compact operators

Colloquium Mathematicae (2015)

• Volume: 138, Issue: 2, page 255-269
• ISSN: 0010-1354

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Abstract

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let ${L}_{w*}\left(E*,F\right)$ (resp. ${K}_{w*}\left(E*,F\right)$) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of ${K}_{w*}\left(E*,F\right)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then ${K}_{w*}\left(E*,F\right)$ has property (wV*). Suppose that ${L}_{w*}\left(E*,F\right)={K}_{w*}\left(E*,F\right)$. Then ${K}_{w*}\left(E*,F\right)$ has property (V*) if and only if E and F have property (V*).

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