Bull. Korean Math. Soc. 2019; 56(3): 563-568
Online first article May 16, 2019 Printed May 31, 2019
https://doi.org/10.4134/BKMS.b170568
Copyright © The Korean Mathematical Society.
Keun Young Lee
Sejong University
This study is concerned with the approximation properties of pairs. For $\lambda \geq 1$, we prove that given a Banach space $X$ and a closed subspace $Z_{0}$, if the pair $(X,Z_{0})$ has the $\lambda$-bounded approximation property ($\lambda$-BAP), then for every ideal $Z$ containing $Z_{0}$, the pair $(Z,Z_{0})$ has the $\lambda$-BAP; further, if $Z$ is a closed subspace of $X$ and the pair $(X,Z)$ has the $\lambda$-BAP, then for every separable subspace $Y_{0}$ of $X$, there exists a separable closed subspace $Y$ containing $Y_{0}$ such that the pair $(Y,Y\cap Z)$ has the $\lambda$-BAP. We also prove that if $Z$ is a separable closed subspace of $X$, then the pair $(X,Z)$ has the $\lambda$-BAP if and only if for every separable subspace $Y_{0}$ of $X$, there exists a separable closed subspace $Y$ containing $Y_{0} \cup Z$ such that the pair $(Y,Z)$ has the $\lambda$-BAP.
Keywords: bounded approximation property of pairs, approximation property of pairs, ideals
MSC numbers: Primary 46B28; Secondary 47L20
Supported by: The author was supported by NRF-2017R1C1B5017026 funded by the Korean Government
2014; 51(3): 667-680
2005; 42(2): 379-386
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd