Cesàro convergent sequences in the Mackey topology

2018/12/25 by José Rodríguez, Rodríguez, José, Jose Israel Rodriguez
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 46B50. Secondary: 47B07

paper · pdf · doi:10.48550/arxiv.1812.10079

openalex publication_date 2018/12/25 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

A Banach space X is said to have property (μ^s) if every weak^*-null sequence in X^* admits a subsequence such that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapień. We prove that property (μ^s) holds for every subspace of a Banach space which is strongly generated by an operator with Banach-Saks adjoint (e.g. a strongly super weakly compactly generated space). The stability of property (μ^s) under ℓ^p-sums is discussed. For a family A of relatively weakly compact subsets of X, we consider the weaker property (μ_A^s) which only requires uniform convergence on the elements of A, and we give some applications to Banach lattices and Lebesgue-Bochner spaces. We show that every Banach lattice with order continuous norm and weak unit has property (μ_A^s) for the family of all L-weakly compact sets. This sharpens a result of de Pagter, Dodds and Sukochev. On the other hand, we prove that L¹(ν,X) (for a finite measure ν) has property (μ_A^s) for the family of all δS-sets whenever X is a subspace of a strongly super weakly compactly generated space.

Related

• Quantifying properties (K) and (μ^s)
• Convex combinations of weak*-convergent sequences and the Mackey topology
• Unbounded continuous operators and unbounded Banach-Saks property in\n Banach lattices
• Some properties on the unbounded absolute weak convergence in Banach lattices
• Observations on some classes of operators on C(K,X)