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

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 $\Phi(s) = c_1 + c_2 2^{-s}$.