On the modulus of cone absolutely summing operators and vector measures of bounded variation

Abstract

Let E and F be Banach lattices. It is shown that if F has the Levi and the Fatou property, then the ordered Banach space 5?1{E,F) of cone absolutely summing operators is a Banach lattice and an order ideal of the Riesz space Sfr{E, F) of regular operators. The same argument yields a Jordan decomposition of F-valued vector measures of bounded variation.

On Connections Between Delta-Convex Mappings and Convex Operators

“…(i) Recall that a Nakano space is a Banach lattice in which every norm-bounded upperdirected set M has a supremum satisfying sup M = sup x∈M x . The particular case of Theorem 3.7 when Y is a Nakano space could also be proved using known results [7,10] on Jordan decomposition of finitely additive vector measures. Such a procedure could also be used in a more general case of a Levi space Y , since we could use Theorem 5.1 below.…”

“…On the other hand, we do not obtain that M (equipped with the norm µ = var(µ, E)) is a normed lattice. If Y is a Nakano space, then M is a Banach lattice (see [10] and [7]). We do not know whether the assumption that Y is σ-Levi is sufficient for some general class of algebras.…”

“…such µ holds in the case when Y is a Nakano space (see [10] and [7]). We observe that the natural decomposition procedure (essentially the same as is sketched in [7]) and a well-known result of [8] on Levi spaces imply that the Jordan decomposition theorem also holds in the more general case when Y is a Levi space.…”