Some Properties of the Second Conjugate of a Tauberian Operator

A bounded linear operator T: XªY Banach spaces.is defined to be Taube- rian provided whenever xn4;X is bounded and T xn.4;Y is weakly conver- gent, then xn4 has a weakly convergent subsequence. Hence, they appear as opposite to weakly compact operators. In 1991 a Tauberian operator T between separable Banach spaces was found such that its second conjugate T** is not Tauberian. Though T** might not be Tauberian, in this paper we prove that it satisfies the following property when X is separable: whenever xUnU4;XUU is bounded and TUU xUnU.4;YUU is weakly convergent, then xUnU4 has a w*- convergent subsequence. Other properties of T** are proved and the nonseparable case is also studied.