CONICAL DECOMPOSITION AND VECTOR LATTICES WITH RESPECT TO SEVERAL PREORDERS

Abstract

The decomposition set-valued mapping in a Banach space $E$ with cones $K_i$, $i = 1, \ldots, n$ describes all decompositions of a given element on addends, such that addend $i$ belongs to the $i$-th cone. We examine the decomposition mapping and its dual.

We study conditions that provide the additivity of the decomposition mapping. For this purpose we introduce and study the Riesz interpolation property and lattice properties of spaces with respect to several preorders. The notion of 2-vector lattice is introduced and studied. Theorems that establish the relationship between the Riesz interpolation property and lattice properties of the dual spaces are given.