Complex Symmetry of Composition Operators on Hilbert Spaces of Entire Dirichlet Series

Abstract

A criterion for boundedness of composition operators acting on a class of Hilbert spaces of entire Dirichlet series, namely the class $\mathcal {H}(E, \beta _{S})$, was obtained in Hou et al. (J. Math. Anal. Appl. 401: 416–429, 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 $\mathcal {H}(E, \beta _{S})$, which may or may not contain constant functions. We then study complex symmetry of composition operators on $\mathcal {H}(E, \beta _{S})$, via analysis of composition conjugations.