Hilbert spaces of entire Dirichlet series and composition operators

The aim of this paper is to introduce Hilbert spaces of entire Dirichlet series with real frequencies and consider composition operators on these spaces. We establish necessary and sufficient conditions for such series to have Ritt order zero, as well as to have finite logarithmic orders. This allows us to apply the Polya theorem on composition of entire functions to consideration of composition operators on the Hilbert spaces of entire Dirichlet series. In particular, criteria for action, boundedness, compactness and compact difference of such operators are obtained.