Uo-convergence and its applications to Cesàro means in Banach lattices

• Published in: Israel journal of mathematics Vol. 220; no. 2; pp. 649 - 689
• Main Authors: Gao, N., Troitsky, V. G., Xanthos, F.
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.06.2017; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Convergence; Group Theory and Generalizations; Lattices (mathematics); Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operators; Theoretical
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-017-1530-y

A net ( x α ) in a vector lattice X is said to uo-converge to x if | x α − x | ∧ u → o 0 for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uoconvergence 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 [27, 26]. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós’ 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 [20, 24, 25].