Composition-Differentiation Operator On Hardy-Hilbert Space of Dirichlet Series

Abstract: In this paper, we establish a compactness criterion for the composition-differentiation operator ( D_\Phi ) in terms of a decay condition of the mean counting function at the boundary of a half-plane. We provide a sufficient condition of the boundedness of the operator ( D_\Phi ) for the symbol ( \Phi ) with zero characteristic. Additionally, we investigate an estimate for the norm of ( D_\Phi ) in the Hardy-Hilbert space of Dirichlet series, specifically with the symbol ( \Phi(s) = c_1 + c_2 2^{-s} ). We also derive an estimate for the approximation numbers of the operator ( D_\Phi ). Moreover, we determine an explicit conditions under which the operator ( D_\Phi ) is self-adjoint and normal. Finally, we describe the spectrum of ( D_\Phi ) when the symbol ( \Phi(s) = c_1 + c_2 2^{-s} ).