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Abstract: A net in a vector lattice X is said to be unbounded order convergent (or uo-convergent, for short) to if the net converges to 0 in order for all . In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X be a Banach lattice. We prove that every norm bounded uo-convergent net in is -convergent iff X has order continuous norm, and that every -convergent net in is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ-order complete Banach lattices the spaces in whose dual space every simultaneously uo- and -convergent sequence converges weakly/in norm. [Copyright &y& Elsevier]

Abstract: A net in a vector lattice X is unbounded order convergent (uo-convergent) to x if for each , and is unbounded order Cauchy (uo-Cauchy) if the net is uo-convergent to 0. In the first part of this article, we study uo-convergent and uo-Cauchy nets in Banach lattices and use them to characterize Banach lattices with the positive Schur property and KB-spaces. In the second part, we use the concept of uo-Cauchy sequences to extend Doob's submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in is almost surely uo-Cauchy in F, where F is an order continuous Banach lattice with a weak unit. [Copyright &y& Elsevier]

The classical Perron–Frobenius theory asserts that an irreducible matrixhas cyclic peripheral spectrum and its spectral radiusis an eigenvalue corresponding to a positive eigenvector. This was extended by Radjavi and Rosenthal to semigroups of matrices and of compact operators on-spaces. We extend this approach to operators on an arbitrary Banach lattice. We prove, in particular, that ifis a commutative irreducible semigroup of positive operators oncontaining a compact operatorthen there exist positive disjoint vectorsinsuch that every operator inacts as a positive scalar multiple of a permutation on. Compactness ofmay be replaced with the assumption thatis peripherally Riesz, i.e. the peripheral spectrum ofis separated from the rest of the spectrum and the corresponding spectral subspaceis finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator, we show thatis a cyclic permutation on,, and iffor someinandthenfor someand. We also extend results of Drnovšek and Levin about peripheral spectra of irreducible operators. [ABSTRACT FROM AUTHOR]