A criterion for boundedness of composition operators acting on a class of Hilbert spaces of entire Dirichlet series, namely the class H(E, βS), was obtained in J. Math. Anal. Appl. ( 2013) for those spaces that do not contain non-zero constant functions, while other possibilities were not studied. In this paper, we first provide a complete characterization of boundedness of composition operators on any space H(E, βS), which may or may not contain constant functions. We then study complex symmetry of composition operators on H(E, βS), via analysis of composition conjugations.

In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a characterization so that the direct sum of the identity plus a weighted forward shift operator on $\ell^2$ is cyclic.

In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some halfplane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a sufficient condition so that the direct sum of the identity plus a weighted forward shift operator on the Hardy space H 2 is cyclic.