Disjointly Homogeneous Banach Lattices and Applications

Abstract

Abstract. This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.

Проблема Мажорации В Банаховых Решетках

“…Некоторые из основных результатов представлены в следующей таблице (более детальную информацию см. в книгах [2], [8] и обзорных статьях [1], [16], [19], [32]). Таблица читается следующим образом: если оператор 𝑆, 𝑇 ∈ ℒ(𝐸, 𝐹 ) таковы, что 0 𝑆 𝑇 и 𝑇 ∈ Φ(𝐸, 𝐹 ), где Φ -какое-нибудь из свойство, перечисленных в первом столбце, то 𝑆 ∈ Φ(𝐸, 𝐹 ), при условии, что 𝐸 и 𝐹 удовлетворяют ограничениям в соответствующей строке второго столбца.…”

A quantitative approach to disjointly non-singular operators

“…The disjointly strictly singular operators (DSS operators) were introduced in [12] as those operators T : E → Y from a Banach lattice E into a Banach space Y such that T is not an isomorphism in any subspace of E generated by a disjoint sequence of non-zero vectors. These operators have been useful in the study of the structure of Banach lattices (see [2], [3] and references therein). More recently, the disjointly non-singular operators (DN-S operators) where introduced in [6] (see also [1]) as those operators T : E → Y that are not strictly singular in any subspace of E generated by a disjoint sequence of non-zero vectors.…”