Operator Valued Series and Vector Valued Multiplier Spaces

‎Let X,Y be normed spaces with L(X,Y) the space of continuous‎ ‎linear operators from X into Y‎. ‎If Tj is a sequence in L(X,Y),‎ ‎the (bounded) multiplier space for the series sumTj is defined to be‎ [ ‎M^{infty}(sum T_{j})={{x_{j}}in l^{infty}(X):sum_{j=1}^{infty}%‎ ‎T_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator S:Minfty(sumTj)rightarrowY associated‎ ‎with the series is defined to be S(xj)=suminftyj=1Tjxj.‎ ‎In the scalar case the summing operator has been used to characterize‎ ‎completeness‎, ‎weakly unconditionall Cauchy series‎, ‎subseries and absolutely‎ ‎convergent series‎. ‎In this paper some of these results are generalized to the‎ ‎case of operator valued series The corresponding space of weak multipliers‎ ‎is also considered.‎