‎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 $s:m^{infty}(sum t_{j})rightarrow y$ associat...

The space of all entire functions represented by vector valued Dirichlet series of two complex variables is considered in this paper. It is equipped with two equivalent topologies. The main result of this paper is concerned with finding the conditions for a base in Xto become a proper base and certain continuous linear operators which are used to determine the proper bases inX.

In the present paper, we study the entire functions represented by vector valued Dirichlet series of several complex variables. The characterizations of their order and type have been obtained. For the sake of simplicity, we have considered the functions of two variables only.

‎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 $S:M^{infty}(sum T_{j})rightarrow Y$ associat...

Spaces of all entire functions f represented by vector valued Dirichlet series and having slow growth have been considered. These are endowed with a certain topology under which they become a Frechet space. On this space the form of linear continuous transformations is characterized. Proper bases have also been characterized in terms of growth parameters.