Family Of Positive Operators Research Articles

Weak convergence in the dual of weak Lp

We consider a Dedekind σ-complete Banach lattice E whose dual is weakly sequentially complete. Suppose that E has a positive element u and a family of positive operators \( \mathcal{G} \) such that (i) each T′, T ∈ \( \mathcal{G} \), is a lattice homomorphism (ii) \( \cup _{T \in \mathcal{G}} \)[−u, u] contains the unit ball of E (iii) for any sequence (xn) ⊂ [0, u] of pairwise disjoint elements and for any sequence (Tn) ⊂ \( \mathcal{G} \) the sequence (Tnxn) is majorized in E.