On the compact approximation property
Tom 160 / 2004
Studia Mathematica 160 (2004), 185-200
MSC: Primary 46B20, 46B28, 47L05.
DOI: 10.4064/sm160-2-6
Streszczenie
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.
