Isometric factorization of weakly compact operators and the approximation property (Q1594898)

The central result of this well written paper is the following new characterization of the approximation property (AP) of a Banach space \(X\): \(X\) has AP iff for any Banach space \(Y\) and any weakly compact operator \(T\in{\mathcal W}(Y, X)\) there is a net \((T_\alpha)\) of finite rank operators \(T_\alpha\in{\mathcal F}(Y, X)\) such that \(\sup_\alpha\|T_\alpha\|\leq\|T\|\) and \(T_\alpha y\to Ty\) for all \(y\in Y\), i.e. the unit ball of \({\mathcal F}(Y, X)\) is dense in the unit ball of \({\mathcal W}(Y, X)\) with respect to the strong operator topology. This result is obtained from a stronger, isometric version of the Davis, Figiel, Johnson and Pełczyński factorization theorem for weakly compact operators. Recall that a linear subspace \(F\) of a Banach space \(E\) is an ideal in \(E\) if \(F^{\perp}\) is the kernel of a norm one projection in \(E^*\). With this notion, a second characterization of the AP of \(X\) and \(X^*\), respectively, can be given: \(X\) has AP iff \({\mathcal F}(Y,X)\) is an ideal in \({\mathcal W}(Y, X)\) for all Banach spaces \(Y\), and, \(X^*\) has AP iff \({\mathcal F}(X, Y)\) is an ideal in \({\mathcal W}(X,Y)\) for all Banach spaces \(Y\). There are several more refinements and generalizations of this result, e.g. a characterization of the compact approximation property of \(X\) can be given by replacing \({\mathcal F}(Y,X)\) in the above criterion by the space \({\mathcal K}(Y, X)\). In the final section it is shown how to get MAP/MCAP from AP/CAP and RNP.