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

On the disjoint weak Banach-Saks operators

“…(4) =⇒ (1) Let (x n ) be a disjoint weakly null sequence of E. By Corollary 3.6 [9], we see that x n uo −→ 0 and hence it follows from the assertion (4) that (T (x n )) has a Cesàro convergent subsequence in Y . That is, T is a disjoint weak Banach-Saks operator.…”

“…(2) =⇒ ( 3) Let (x n ) be a weakly null sequence of E + . Since E is order continuous, then it follows from Proposition 4.5 [9] and Proposition 4.7 of [9] that x n k uo −→ 0 for some subsequence (x n k ). The assertion (2) yields (T (x n k )) has a subsequence whose Cesàro sequence is norm convergent in F .…”

“…An order continuous Banach lattice E is said to have the subsequence splitting property if for any norm bounded sequence (x n ) there exist a subsequence (x n k ) of (x n ) and two sequences (y k ) and (z k ) such that x n k = y k + z k , (y k ) is almost order bounded, (z k ) is pairwise disjoint and y k ⊥ z k for all k (see [9]).…”

“…We mention that order convergence implies uo-convergence and they coincide for order bounded nets. We note that every disjoint net is uo-null (see [9]).…”

“…Consequently, we give a new characterization of the disjoint weak Banach-Saks property. Also, we give a generalization of Proposition 6.9 [9] and of Propositions 6.15 [9] . Furthermore, we study the relationship between this class of operators and that of weak Banach-Saks operators (resp; almost Banach-Saks operators, order weakly compact operators and weakly compact operators).…”