Composition Operators On Entire Dirichlet Series With Real Frequencies

Dirichlet series with real frequencies, or generalized Dirichlet series, extend the classical Dirichlet series, which are widely used in number theory and complex analysis. In recent years, composition operators — situated at the intersection of complex analysis, functional analysis, and operator theory — have been extensively studied on various function spaces, including spaces of classical Dirichlet series.In functional analysis, the study of Hilbert and Banach spaces of Dirichlet series, and the action of composition operators on these spaces, has garnered significant attention. This includes exploring properties such as boundedness, compactness, essential norms, Fredholmness, and cyclicity, particularly on entire Dirichlet series with real frequencies.Given the two directions of development in the study of the Dirichlet series — either holomorphic on the half-plane or entire across the whole plane — this monograph addresses the pressing need for a comprehensive investigation on composition operators acting on entire generalized Dirichlet series. It provides foundational knowledge, recent developments, and several open questions for further research, offering a valuable resource for both established researchers and those new to the field.