On the compact approximation property

Vegard Lima¹ ; Åsvald Lima¹ ; Olav Nygaard¹

¹ Department of Mathematics Agder University College Serviceboks 422 4604 Kristiansand, Norway

Studia Mathematica, Tome 160 (2004) no. 2, pp. 185-200

Résumé

We show that a Banach space $X$ has the compact approximation property if and only if for every Banach space $Y$ and every weakly compact operator $T : Y \rightarrow X$, the space $$ \mathfrak{E} = \{ S \circ T : S\ \hbox{compact operator on}\ X \} $$ is an ideal in $\mathfrak{F} = \mathop{\rm span}(\mathfrak{E},\{T\})$ if and only if for every Banach space $Y$ and every weakly compact operator $T: Y \rightarrow X$, there is a net $(S_\gamma)$ of compact operators on $X$ such that $\sup_\gamma \|S_\gamma T\| \le \|T\|$ and $S_\gamma \rightarrow I_X$ in the strong operator topology. Similar results for dual spaces are also proved.

Détail

DOI : 10.4064/sm160-2-6

Keywords: banach space has compact approximation property only every banach space every weakly compact operator rightarrow space mathfrak circ hbox compact operator ideal mathfrak mathop span mathfrak only every banach space every weakly compact operator rightarrow there net gamma compact operators sup gamma gamma gamma rightarrow strong operator topology similar results dual spaces proved