Domination problem on Banach lattices and almost weak compactness

Abstract

In this paper we give several new results concerning domination problem in the setting of positive operators between Banach lattices. Mainly, it is proved that every positive operator R on a Banach lattice E dominated by an almost weakly compact operator T satisfies that the R 2 is almost weakly compact. Domination by strictly singular operators is also considered. Moreover, we present some interesting connections between strictly singular, disjointly strictly singular and almost weakly compact operators.

Complete latticeability in vector‐valued sequence spaces

“…In Remark 2.3, we shall explain why we cannot go to a space smaller than 𝑐 𝑤 0 (𝐸). The positive Schur property in Banach lattices (positive-or, equivalently, disjoint-weakly null sequences are norm null) was introduced by Wnuk [41] and Räbiger [39] and it has been extensively studied, for some recent developments see [4,8,11,22,40,43,44]. Question (2) of the Introduction is nothing but the natural lattice counterpart of the problem solved by Jiménez-Rodríguez in the Banach space setting.…”

The positive polynomial Schur property in Banach lattices

“…In the setting of Banach lattices, the positive Schur property (weakly null sequences formed by positive vectors are norm null) has been extensively studied, recent developments can be found in [5,8,10,16,33,36,37]. So it is a natural step to consider the lattice counterpart of the polynomial Schur property, which is the main subject of this paper.…”

Complete latticeability in vector-valued sequence spaces

“…The positive Schur property in Banach lattices (positive -or, equivalently, disjointweakly null sequences are norm null) was introduced by Wnuk [29] and Räbiger [27] and has been extensively studied, for some recent developments see [4,8,9,14,28,31,32]. Oikhberg [25] coined the following terms: a subset A of a Banach lattice is latticeable (completely latticeable) if there exists a (closed) infinite dimensional sublattice of E all of whose elements but the origin belong to A (see also [26]).…”