Disjointly homogeneous Banach lattices and compact products of operators

Abstract

Keywords:Positive operator Disjointly strictly singular operator AM-compact operator Dunford-Pettis operator The notion of disjointly homogeneous Banach lattice is introduced. In these spaces every two disjoint sequences share equivalent subsequences. It is proved that on this class of Banach lattices the product of a regular AM-compact and a regular disjointly strictly singular operators is always a compact operator.

Disjointly Homogeneous Banach Lattices and Applications

“…Tsirelson space also falls within the category of DH Banach lattices, as shown in [15]. As a consequence, we deduce that DH Banach lattices need not be p-DH for any 1 p ∞.…”

“…The notion of disjointly homogeneous Banach lattice was first introduced in [15]; let us recall its definition. Definition 2.1.…”

“…The motivation which gave rise to the definition of a disjointly homogeneous space was to decide the compactness of the iterations of a given operator which already enjoyed nice close-to-compactness properties. Thus the first and third authors together with V. G. Troitsky considered this notion for the first time in [15].…”

Duality problem for disjointly homogeneous rearrangement invariant spaces

“…is a bounded projection in W . In fact, applying successively inequalities (14), (18) and (16), for any x ∈ W we obtain…”

“…In particular, given 1 ≤ p ≤ ∞, a Banach lattice E is p-disjointly homogeneous (shortly p-DH) if each normalized disjoint sequence in E has a subsequence equivalent to the unit vector basis of l p (c 0 when p = ∞). These notions were first introduced in [14] and proved to be very useful in studying the general problem of identifying Banach lattices E such that the ideals of strictly singular and compact operators bounded in E coincide [11] (see also survey [13] and references therein). Results obtained there can be treated as a continuation and development of a classical theorem of V. D. Milman [33] which states that every strictly singular operator in L p (µ) has compact square.…”