Uo-convergence and its applications to Cesàro means in Banach lattices
By Niushan Gao, V. G. Troitsky, Foivos Xanthos. 2015
174 citations recorded by the source corpus.
A net (xα) in a vector lattice X is said to uo-converge to x if $$\left| {{x_\alpha } - x} \right| \wedge u\xrightarrow{o}0$$|xα−x|∧u→o0 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].
Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/a7cce840d5b191e24bff0c424c824419d80598fe
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Theory and Mathematics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Unbounded norm topology in Banach lattices (2016, background): “corresponds to 1, L∞(µ) is a norm-dense ideal in X, and both inclusions in L∞(µ) ⊆ X⊆ L1(µ) are continuous. We call L1(µ) an AL-representation for Xand e. Let (xn) be a sequence in X. It was shown in [GTX] that xn −→uo 0 in Xiff xn −−→a.e. 0 in L 1(µ); it was shown in [DOT] that xn −→un 0 in Xiff xn →”