Research Article

The (E, E)-approximation property for Banach spaces

Published in: Quaestiones Mathematicae
Volume 46 , issue 10 , 2023 , pages: 2147–2161

DOI: 10.2989/16073606.2022.2143450

Author(s): Ju Myung Kim Sejong University, Korea , Bentuo Zheng The University of Memphis, USA

Keywords: Primary: 46B28 , Secondary: 47L05 , Banach space , unconditional basis , approximation property

Abstract

For a given Banach space E with an 1-unconditional basis, we introduce the (E, E)-approximation property ((E, E)-AP) which is a natural generalization of the classical approximation property (AP) and the (p, p)-AP. It is shown that the AP always implies the (E, E)-AP. If E has an 1-unconditional and shrinking basis, then the (E*, E*)-AP for X* implies the (E, E)-AP for X. If in addition that the unconditional basis for E is subsymmetric, then X* has the (E*, E*)-AP implies that X has the duality (E, E)-AP.

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