Order continuous extensions of positive compact operators on Banach lattices

2011/02/24 by Jin Xi Chen, Chen, Jin Xi, Zi Li Chen +4
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA

paper · pdf · doi:10.48550/arxiv.1102.4912

7 pages

arxiv created 2011/02/24 · openalex publication_date 2011/02/24 · arxiv updated 2011/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E and F be Banach lattices. Let G be a vector sublattice of E and T: G→ 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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