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 ${T_{j}}$ is a sequence in $L(X,Y)$,‎ ‎the (bounded) multiplier space for the series $sum T_{j}$ 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 $S:M^{infty}(sum T_{j})rightarrow Y$ associated‎ ‎with the series is defined to be $S({x_{j}})=sum_{j=1}^{infty}T_{j}x_{j}$.‎ ‎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.‎