A note on weak almost limited operators

Abstract

Let us recall that an operator T : E → F, between two Banach lattices, is said to be weak* Dunford-Pettis (resp. weak almost limited) if fn (T xn) → 0 whenever (xn) converges weakly to 0 in E and (fn) converges weak* to 0 in F (resp. fn (T xn) → 0 for all weakly null sequences (xn) ⊂ E and all weak* null sequences (fn) ⊂ F with pairwise disjoint terms). In this note, we state some sufficient conditions for an operator R : G → E(resp. S : F → G), between Banach lattices, under which the product T R (resp. ST) is weak* Dunford-Pettis whenever T : E → F is an order bounded weak almost limited operator. As a consequence, we establish the coincidence of the above two classes of operators on order bounded operators, under a suitable lattice operations' sequential continuity of the spaces (resp. their duals) between which the operators are defined. We also look at the order structure of the vector space of weak almost limited operators between Banach lattices.

The Grothendieck property from an ordered point of view

“…The second part of the definition was initially defined in [4]. The third part was proposed in [2] at first. Parts (iv) and (v) are new definitions using unbounded convergence and disjointness.…”

Grothendieck-type subsets of Banach lattices

“…Next proposition is a small improvement of Proposition 4.10 in [15]. Proposition 2.7 If E ′ , the dual of some Banach lattice E, does not contain any copy of ℓ 1 , then E has the weak Grothendieck property.…”

“…✷ In [15], the authors proved that every L-space has the weak Grothendieck property. Also in [14], it is proved that the Banach lattice ( n∈N ℓ n…”

“…every positive weak* null sequence in E ′ is weakly null ). We refer to [21] and [15] for definitions and some results concerning those two properties.…”

A note on the Banach lattice $c_0( \ell_2^n)$, its dual and its bidual

“…Following [11], E has the weak Grothendieck property if every disjoint weak* null sequence in E ′ is weakly null. Clearly, the Grothendieck property implies both the positive Grothendick and the weak Grothendieck.…”