Ideal properties of regular operators between Banach lattices

Domination problem on Banach lattices and almost weak compactness

“…Each subset B ⊂ E is contained in a smallest solid subset Sol(B) of E containing B, called the solid hull of B. We also recall that a positive element u ∈ E is said to be a quasi-interior point whenever the ideal generated by u, E u = {v ∈ E; ∃λ > 0 with | v |≤ λu} is norm dense in E. N. Kalton and P. Saab proved in [16] the following approximation result Theorem 10 Let E and F be Banach lattices each with a quasi-interior positive element. Let T be a positive operator T : E → F and consider A ⊂ E and B ⊂ F two solid bounded sets.…”

Charalambos D. Aliprantis (1946–2009)

“…Then Kalton and Saab [21] showed that it suffices that F alone have an order continuous norm. Subsequently, Aliprantis and Burkinshaw included as Theorem 19.10 of [11] a proof that if the lattice operations in E are weakly sequentially continuous then the domination property also holds, whilst the current author showed in [32] that the two conditions stated here are the only possible ones.…”

Domination problems for strictly singular operators and other related classes

“…These were later improved by Kalton and Saab [33] showing that if F is order continuous and T is Dunford-Pettis then R is also Dunford-Pettis (see [56] in this volume for more comments on the history of these results). Note also that converse domination results have been given in [8,54,55].…”

“…Moreover, the power problem for the class of Dunford-Pettis operators was also studied by Kalton and Saab in [33]: If T is Dunford-Pettis, then R 2 is also Dunford-Pettis.…”