Order continuous extensions of positive compact operators on Banach lattices

Details Order continuous extensions of positive compact operators on Banach lattices Order continuous extensions of positive compact operators on Banach lattices

2011-02-24

arxiv.org/abs/1102.4912v1

Mathematics

Functional Analysis

7 pages

Scientific paper

Let $E$ and $F$ be Banach lattices. Let $G$ be a vector sublattice of $E$ and $T: G\rightarrow F$ be an order continuous positive compact (resp. weakly compact) operators. We show that if $G$ is an ideal or an order dense sublattice of $E$, then $T$ has a norm preserving compact (resp. weakly compact) positive extension to $E$ which is likewise order continuous on $E$. In particular, we prove that every compact positive orthomorphism on an order dense sublattice of $E$ extends uniquely to a compact positive orthomorphism on $E$.

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