The property (d) and the almost limited completely continuous operators

Grothendieck-type subsets of Banach lattices

“…Now assume that |A| is almost Grothendieck and let T : E → c 0 be a disjoint operator. By Theorem 2.7 of [13], T is a regular, hence, order bounded operator. Lemma 2.10 yields that |T | is also a disjoint operator.…”

“…It is easy to see that every Banach lattice with the weak Grothendieck property has the property (d). Theorem 2.7 in [13] shows that E has the property (d) if and only if every disjoint linear operator on E is regular, i.e. it can be written as the sum of two positive operators.…”

“…Assume that |A| is a Grothendieck set and let T : E → c 0 be a bounded linear operator. Since the lattice operations in E ′ are weak* sequentially continuous, T is a regular operator [The same proof used in[13, Theorem 2.7] holds in this case]. Now we can proceed with the same argument as in Theorem 2.11 in order to get that sol(A) is a Grothendieck set.…”