Dual Results of Factorization for Operators

We study the duality properties of the well-known DFJP factorization of operators 3] by means of a reenement of it. Given an operator T : X ! Y we consider a decomposition T = jU k , where U : E ! F is an isomorphism, and j , U k are the factors in the DFJP factorization. If T is the conjugate operator of T , and T : X =X ! Y =Y is the operator given by T (x + X) := T x + Y (x 2 X), then we show that the decompositions of T and T are precisely k U j = (jUk) and j U k. From this result we derive several consequences. For example, we detect new operator ideals with the factorization property, we characterize operators whose conjugate is Rosenthal, and using a result of Valdivia 11] we show that an operator T such that T has separable range can be decomposed as T = S + K , where S has separable range and K is weakly compact. 0. Introduction For a (continuous linear) operator T 2 L(X; Y) we shall introduce a decomposition T = jUk in which U is an isomorphism, j is an injective tauberian operator, and k is a cotauberian operator with dense range. This decomposition is inspired by the equivalent versions of the real interpolation method of Banach spaces 2], and it is a reenement of the well-known DFJP factorization introduced in 3] which factorizes T in two factors: j and Uk. Moreover, the factorization of T in two factors jU and k was considered in 5]. We show that k U j coincides with the decomposition of the conjugate operator T 2 L(Y ; X), and j U k coincides with the decomposition of the operator T 2 L(X =X; Y =Y). Moreover, if T belongs to a closed operator ideal A , then k and j belong to the injective hull and the surjective hull of A , respectively. In this way the decomposition of an operator makes clear the duality properties and the symmetry of the DFJP factorization. As an application we obtain necessary conditions for the factorization property for an operator ideal, and we show that some operator ideals deened in terms of T or T verify this property. Also we characterize the class of operators whose