Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pelczynski factorization of weakly compact operators, we prove that a Banach space X has the approximation property if and only if, for every Banach space Y, the finite rank operators of norm less than or equal to 1 are dense in the unit ball of W(Y,X), the space of weakly compact operators from Y to X, in the strong operator topology. We also show that, for every finite dimensional subspace F of W(Y, X), there are a reflexive space Z, a norm one operator J: Y --> Z, and an isometry Phi: F --> W(Z, X) which preserves finite rank and compact operators so that T = Phi (T) circle J for all T is an element of F. This enables us to prove that X has the approximation property if and only if the finite rank operators form an ideal in W(Y, X) for all Banach spaces Y.