Spaces of analytic functions represented by Dirichlet series of two complex variables

Abstract

We consider the space X of all analytic functions $$f(s_1 ,s_2 ) = \sum\limits_{m, n - 1}^\infty {a_{mn} exp} (s_1 \lambda _m + s_2 \mu _n )$$ of two complex variables s1 and s2, equipping it with the natural locally convex topology and using the growth parameter, the order of f as defined recently by the authors. Under this topology X becomes a Frechet space. Apart from finding the characterization of continuous linear functionals, linear transformation on X, we have obtained the necessary and sufficient conditions for a double sequence in X to be a proper bases.