2020/07/11 by Omid Zabeti, Zabeti, Omid
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2007.05734
openalex publication_date 2020/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the equivalent definition of a continuous operator between\nBanach spaces in terms of weakly null nets, we introduce unbounded continuous\noperators by replacing weak convergence with the unbounded absolutely weak\nconvergence ( uaw-convergence) in the definition of a continuous operator\nbetween Banach lattices. We characterize order continuous Banach lattices and\nreflexive Banach lattices in terms of these spaces of operators. Moreover,\nmotivated by characterizing of a reflexive Banach lattice in terms of unbounded\nabsolutely weakly Cauchy sequences, we consider pre-unbounded operators between\nBanach lattices which maps uaw-Cauchy sequences to weakly ( uaw- or norm)\nconvergent sequences. This allows us to characterize KB-spaces and reflexive\nspaces in terms of these operators, too. Furthermore, we consider the unbounded\nBanach-Saks property as an unbounded version of the weak Banach-Saks property.\nThere are many considerable relations between spaces possessing the unbounded\nBanach-Saks property with spaces fulfilled by different types of the known\nBanach-Saks property. In particular, we characterize order continuous Banach\nlattices in terms of these relations, as well.\n