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Dual results of factorization for operators. (English) Zbl 0795.46013

We study the duality properties of the well-known DFJP factorization of operators [W.J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski, J. Funct. Anal. 17, 311-327 (1974; Zbl 0306.46020)] by means of a refinement of it. Given an operator \(T: X\to Y\) we consider a decomposition \(T= jUk\), where \(U: E\to F\) is an isomorphism, and \(j\), \(Uk\) are the factors in the DFJP factorization.
If \(T^*\) is the conjugate operator of \(T\), and \(\overline{T}: X^{**}/X\to Y^{**}/Y\) is the operator given by \(\overline{T}(x+X):= T^{**} x+Y\) \((x\in X^{**})\), then we show that the decompositions of \(T^*\) and \(\overline{T}\) are precisely \(k^* U^* j^*= (jUk)^*\) and \(\overline{j} \overline{U} \overline{k}\). 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 M. Valdivia [Stud. Math. 60, 11-13 (1977; Zbl 0354.46012)] we show that an operator \(T\) such that \(\overline{T}\) has separable range can be decomposed as \(T=S+K\), where \(S^{**}\) has separable range and \(K\) is weakly compact.

MSC:

46B70 Interpolation between normed linear spaces
47L20 Operator ideals
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
46B10 Duality and reflexivity in normed linear and Banach spaces
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