Analytic reproducing kernel Hilbert spaces and their operators

Abstract

A criterion for boundedness of composition operators acting on the general class of Hilbert spaces of entire Dirichlet series, namely the class $\mathcal{H}(\beta,E)$, was obtained in [15]. Varied results of properties were analysed in earlier papers [22, 13, 2]. In this thesis we extend these results to the general setting of spaces of Dirichlet series holomorphic on the half-plane. A complete characterisation of boundedness of polynomial-induced composition operators is found. We then study several properties of these operators, obtaining several characterisations in complex symmetry, compactness, etc. A proof that a system of normalised reproducing kernels $(\widetilde{k_{\lambda_n}})$ is never a frame for the Hardy space $H^2$ is also analysed. A generalisation of the method was made to determine classes of spaces and sequences $(\widetilde{k_{\lambda_n}})$ which do not constitute frames for their parent spaces.