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Cesàro Convergent Sequences in the Mackey Topology

  • José RodríguezEmail author
Article
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Abstract

A Banach space X is said to have property (\(\mu ^s\)) 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 \((\mu ^s)\) 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 \((\mu ^s)\) under \(\ell ^p\)-sums is discussed. For a family \(\mathcal {A}\) of relatively weakly compact subsets of X, we consider the weaker property \((\mu _\mathcal {A}^s)\) which only requires uniform convergence on the elements of \(\mathcal {A}\), 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 \((\mu _\mathcal {A}^s)\) 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 \(L^1(\nu ,X)\) (for a finite measure \(\nu \)) has property \((\mu _\mathcal {A}^s)\) for the family of all \(\delta \mathcal {S}\)-sets whenever X is a subspace of a strongly super weakly compactly generated space.

Keywords

Mackey topology Cesàro convergence Banach–Saks property strongly super weakly compactly generated space Lebesgue–Bochner space 

Mathematics Subject Classification

Primary 46B50 Secondary 47B07 

Notes

Acknowledgements

This research was supported by projects MTM2014-54182-P and MTM2017-86182-P (AEI/FEDER, UE).

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dpto. de Ingeniería y Tecnología de Computadores Facultad de InformáticaUniversidad de MurciaMurciaSpain

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