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EQUIVALENCE OF SEMI-NORMS RELATED TO SUPER WEAKLY COMPACT OPERATORS

Part of: General theory of linear operators Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  22 June 2021

KUN TU Open the ORCID record for KUN TU [Opens in a new window]
KUN TU*
Affiliation:
School of Mathematical Sciences, Yangzhou University, Siwangting Road No. 180, Yangzhou 225002, Jiangsu, China
*
e-mail: tukun@yzu.edu.cn
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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.

Keywords

measure of super weak noncompactnesssuper weakly compact approximationsuper weak compact operators

MSC classification

Primary: 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 506 - 518
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Supported in part by NSFC, grant no. 11701501, and funding from Yangzhou University.

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