Lifting some approximation properties from a dual space $X’$ to the Banach space $X$
Volume 257 / 2021
Studia Mathematica 257 (2021), 287-294
MSC: Primary 46B28, 47L20; Secondary 46B45.
DOI: 10.4064/sm190826-20-2
Published online: 5 October 2020
Abstract
For a fixed Banach operator ideal $\mathcal A $, we characterize $\mathcal A $-compact sets (in the sense of Carl and Stephani) that are determined by $c_0$ via the Banach composition ideal $\mathcal A \circ \mathfrak K_{\infty }$, with $\mathfrak K_{\infty }$ the Banach ideal of Fourie and Swart. This characterization allows us to relate $\mathcal K _{\mathcal A} $-approximation properties on a Banach space and $\mathcal K _{\mathcal B} $-approximation properties on its dual space, where $\mathcal A $ and $\mathcal B $ are ideals linked by some classical procedures. These approximation properties have been widely studied in several papers in the last years.
