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arXiv · 1509.07914

Uo-convergence and its applications to Ces\`aro means in Banach lattices

Niushan Gao·Vladimir G. Troitsky·Foivos Xanthos
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Abstract

A net $(x_\alpha)$ in a vector lattice $X$ is said to uo-converge to $x$ if $|x_\alpha-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Ces\`aro means in Banach lattices. In particular, we establish an intrinsic version of Koml\'os' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].

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Niushan Gao, Vladimir G. Troitsky, Foivos Xanthos. 2015-09-25. Uo-convergence and its applications to Ces\`aro means in Banach lattices. https://arxiv.org/abs/1509.07914

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