The K_A and the K_A uniform approximation property

SILVIA LASSALLE; PABLO TURCO

Tartu

Congreso; Kangro 100- Methods of Analysis and Algebra; 2013

In 1984, Carl and Stephani defined, for a fixed Banach operator ideal $\mathcal A$, the notion of $\mathcal A$-compact sets and the operator ideal of $\mathcal A$-compact operators, denoted by $\mathcal K_{\mathcal A}$. We use the Carl and Stephani theory to inspect two types of approximation properties. The first is rather standard. We say that a Banach space $E$ has the $\mathcal K_\mathcal A$-uniform approximation property if the identity map is uniformly approximated by finite rank operators on $\mathcal A$-compact sets. For the second one, we introduce a way to measure the size of $\mathcal A$-compact sets and use it to give a norm to $\mathcal K_{\mathcal A}$. The geometric results obtained for $\mathcal K_{\mathcal A}$ are applied to give different characterizations of the $\mathcal K_{\mathcal A}$-approximation property, definded by Oja and the authors independently. This approach allow us to undertake the study of both approximation property in tandem. In particular, when $\mathcal A=\mathcal N^p$ the ideal of right $p$-nuclear operators, we cover the $p$-approximation property and the $\kappa_p$-approximation property, which were studied in the past 10 years by several authors.