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SOME RESULTS OF THE $\mathcal K_{\mathcal A}$-APPROXIMATION PROPERTY FOR BANACH SPACES

Part of: Linear spaces and algebras of operators Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 August 2018

JU MYUNG KIM
JU MYUNG KIM*
Affiliation:
Department of Mathematics, Sejong University, Seoul 05006, South Korea e-mail: kjm21@sejong.ac.kr
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Abstract

Given a Banach operator ideal $\mathcal A$, we investigate the approximation property related to the ideal of $\mathcal A$-compact operators, $\mathcal K_{\mathcal A}$-AP. We prove that a Banach space X has the $\mathcal K_{\mathcal A}$-AP if and only if there exists a λ ≥ 1 such that for every Banach space Y and every R ∈ $\mathcal K_{\mathcal A}$(Y, X),

$$ \begin{equation} R \in \overline {\{SR : S \in \mathcal F(X, X), \|SR\|_{\mathcal K_{\mathcal A}} \leq \lambda \|R\|_{\mathcal K_{\mathcal A}}\}}^{\tau_{c}}. \end{equation} $$
For a surjective, maximal and right-accessible Banach operator ideal $\mathcal A$, we prove that a Banach space X has the $\mathcal K_{(\mathcal A^{{\rm adj}})^{{\rm dual}}}$-AP if the dual space of X has the $\mathcal K_{\mathcal A}$-AP.

MSC classification

Primary: 46B28: Spaces of operators; tensor products; approximation properties
Secondary: 46B45: Banach sequence spaces 47L20: Operator ideals
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 545 - 555
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

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