Measure of super weak noncompactness in some Banach sequence spaces

Tu, Kun

doi:10.1016/j.jmaa.2020.124813

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2021

2022

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Cited by 2 publications

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“…A subset A of a Banach space X is relatively super weakly compact if and only if A U is relatively weakly compact for any free ultrafilter U. Equivalently, A is relatively super weakly compact if and only if, for every ε > 0, there is a super weakly compact set S ⊂ X such that A ⊂ S + εB X . For more details on super weakly compact sets, see [5,6,16,17,20].…”

Section: Introductionmentioning

confidence: 99%

Equivalence of Semi-Norms Related to Super Weakly Compact Operators

2021

Bull. Aust. Math. Soc.

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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.

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Section: Introductionmentioning

confidence: 99%