Equivalence of Semi-Norms Related to Super Weakly Compact Operators

Abstract

We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

Applications of the quantification of super weak compactness

“…Using γ 2 as a measure of weak noncompactness for sets and operators, the quantified version of Gantmacher theorem [1] would become an equality. De Blasi's measure applied to operators does not satisfy a similar quantified Gantmacher result, as observed in [1] after an example from [4], neither does the measure on super weak noncompactness introduced by Tu [48], inspired by De Blasi's definition, as…”

“…The class of SWC sets lies strictly between the norm compact and the weakly compact subsets. The theory of SWC sets has been developed during the last 15 years in a series of papers [43,14,15,44,50,16,47,48,38,30].…”