Extension of order bounded operators

2019/05/26 by Kazem Haghnejad Azar, Azar, Kazem Haghnejad
Computer Science · Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1905.10721

openalex publication_date 2019/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assume that a normed lattice E is order dense majorizing of a vector lattice E^t. There is an extension norm \Vert.\Vert_t for E^t and we extend some lattice and topological properties of normed lattice (E,\Vert.\Vert) to new normed lattice (E^t,\Vert.\Vert_t). For a Dedekind complete Banach lattice F, T^t is an extension of T from E^t into F whenever T is an order bounded operator from E into F. For each positive operator T, we have \Vert T\Vert=\Vert T^t\Vert and we show that T^t is a lattice homomorphism from E^t into F and moreover T^t∈ L_n(E^t,F) whenever 0≤ T∈ L_n(E,F) and T(x\wedge y)=Tx \wedge Ty for each 0≤ x,y∈ E. We also extend some lattice and topological properties of T∈ L_b(E,F) to the extension operator T^t∈ L_b(E^t,F).

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