Classes of strictly singular operators and their products

Abstract

221 222 G. ANDROULAKIS ET AL. Isr. J. Math. ABSTRACT V. D. Milman proved in [20] that the product of two strictly singular operators on Lp[0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ -strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of S ξ -hereditarily indecomposable Banach spaces and we examine the operators on them.

Disjointly homogeneous Banach lattices and compact products of operators

“…The following result improves the ones obtained in [2,10,16] in the setting of disjointly homogeneous Banach lattices.…”

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

“…Other instances where certain notions where quantified by using the Schreier families (S ξ ) ξ<ω 1 , are the S ξ -unconditional basic sequences ( [6], [13]), S ξ -convex combinations of sequences in Banach spaces There are two main parts in this article. The first is an extension of a result on strictly singular operators by the first named author, P. Dodos, G. Sirotkin and V. Troitsky, ( [2]). …”

Finitely strictly singular operators between James spaces

“…Some partial results in this direction were obtained in [1,11]. We answer this question in the negative by showing that the operator in [12] is, in fact, finitely strictly singular.…”

“…Actually, each property defines an operator ideal. We refer the reader to [1,7,[9][10][11]13] for more information about strictly and finitely strictly singular operators. All the Banach spaces in this paper are assumed to be over real scalars.…”