Descriptive Set Theoretic Methods Applied to Strictly Singular and Strictly Cosingular Operators

Abstract

Abstract. The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of ω1 subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied.

Quantitative Bessaga–Pełczyński property and quantitative Rosenthal property

“…Since X has the Bessaga–Pełczyński property, the operator is not strictly singular and hence is also not strictly singular. It follows from [, Theorem 3.1] that T is not strictly cosingular. Hence, has the Pełczyński property.…”

“…Remark The added assumption that is separable is inspired by [, Theorem 3.1]. We do not know whether there are other appropriate conditions or even the condition that is separable can be removed.…”

Genericity and Universality for Operator Ideals

“…Denote the subclass of S ξ -strictly singular operators by S ξ -S. It is clear that S 0 -S coincides with the class of compact operators. These classes where defined in [3] and have been extensively studied [2,11,14,48]. In particular, while it is shown that (S ξ -S) ξ<ω 1 separably refines S. In general, S ξ -S need not satisfy the additive property of being an operator ideal [45].…”

An ordinal indexing on the space of strictly singular operators

“…Presently, the two most famous examples are the Szlenk and Bourgain indexes [12,19] (see [17] for a excellent exposition of both). More recently, the Schreier families (S ξ ) ξ<ω 1 (introduced in [1]) have been used to index subclasses of classes of separable Banach spaces and classes of operators between separable Banach spaces [2,3].…”