A double-dual characterization of Rosenthal and semitauberian operators

Abstract

Tauberian operators between Banach spaces were introduced by N. Kalton and A. Wilansky [Proc. Amer. Math. Soc. 57 (1976), no. 2, 251–255;] as those operators T:X→Y whose second conjugate satisfies (T **) −1 Y⊂X . Note that ker(T **)⊂X is a necessary, but not sufficient, condition for T to be Tauberian. These operators were characterized by the "preservation'' of weakly convergent sequences: a bounded sequence (x n ) in X admits a weakly convergent subsequence whenever (Tx n ) is weakly convergent. This is a key fact in the study of the factorization due to W. J. Davis et al. [J. Functional Analysis 17 (1974), 311–327;], which allows one to factorize every weakly compact operator through a reflexive space. Semi-Tauberian operators are defined as those operators that preserve weakly Cauchy sequences, and Rosenthal operators are defined by the opposite property: they take bounded sequences into sequences admitting weakly Cauchy subsequences. The class of semi-Tauberian operators is formally similar to that of Tauberian operators. For example, it was shown by the reviewer and V. M. Onieva [Proc. Amer. Math. Soc. 108 (1990), no. 2, 399–405;] that T is Tauberian [semi-Tauberian] if and only if for every compact operator K the kernel ker(T+K) is reflexive [contains no copy of l 1 ]. In the present paper several characterizations of semi-Tauberian operators and Rosenthal operators are presented. The most remarkable result emphasizes the differences with Tauberian operators: In the case X separable, T:X→Y is semi-Tauberian if and only if ker(T **) is contained in the class B 1 (X) of vectors in X ∗∗ which are limits of weakly Cauchy sequences in X ; and this is equivalent to either (T **) −1 B 1 (Y)⊂B 1 (X) , or (T **) −1 B 1 (Y)⊂X . Analogous results are proved in the case X non-separable, in terms of suitable classes of vectors in X **.