Abstract
We show that a Banach spaces X has the compact approx- imation property if and only if for every Banach space Y and every weakly compact operator T : Y ! X, the space E = {S T : S compact operator on X} is an ideal in F = span(E,{T}) if and only if for every Banach space Y and every weakly compact operator T : Y ! X, there is a net (S ) of compact operators on X such that sup kS Tk k Tk and S ! IX in the strong operator topology. Similar results for dual spaces are also proved.
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