# 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*).

## How to cite

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Ioana Ghenciu. "Weak precompactness and property (V*) in spaces of compact operators." Colloquium Mathematicae 138.2 (2015): 255-269. <http://eudml.org/doc/283616>.

@article{IoanaGhenciu2015,
abstract = {We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_\{w*\}(E*,F)$ (resp. $K_\{w*\}(E*,F)$) 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 $K_\{w*\}(E*,F)$ 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 $K_\{w*\}(E*,F)$ has property (wV*). Suppose that $L_\{w*\}(E*,F) = K_\{w*\}(E*,F)$. Then $K_\{w*\}(E*,F)$ has property (V*) if and only if E and F have property (V*).},
author = {Ioana Ghenciu},
journal = {Colloquium Mathematicae},
keywords = {weakly precompact sets; spaces of operators; compact operators; property },
language = {eng},
number = {2},
pages = {255-269},
title = {Weak precompactness and property (V*) in spaces of compact operators},
url = {http://eudml.org/doc/283616},
volume = {138},
year = {2015},
}

TY - JOUR
AU - Ioana Ghenciu
TI - Weak precompactness and property (V*) in spaces of compact operators
JO - Colloquium Mathematicae
PY - 2015
VL - 138
IS - 2
SP - 255
EP - 269
AB - We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w*}(E*,F)$ (resp. $K_{w*}(E*,F)$) 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 $K_{w*}(E*,F)$ 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 $K_{w*}(E*,F)$ has property (wV*). Suppose that $L_{w*}(E*,F) = K_{w*}(E*,F)$. Then $K_{w*}(E*,F)$ has property (V*) if and only if E and F have property (V*).
LA - eng
KW - weakly precompact sets; spaces of operators; compact operators; property
UR - http://eudml.org/doc/283616
ER -

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