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

In this paper, we establish a compactness criterion for the composition-differentiation operator $D_{\mathrm{\Phi }}$ 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 $D_{\mathrm{\Phi }}$ for the symbol $\mathrm{\Phi }$ with zero characteristic. Additionally, we investigate an estimate for the norm of $D_{\mathrm{\Phi }}$ in the Hardy-Hilbert space of Dirichlet series, specifically with the symbol $\mathrm{\Phi }(s)=c_1+c_2{2}^{-s}$. We also derive an estimate for the approximation numbers of the operator $D_{\mathrm{\Phi }}$. Moreover, we determine an explicit conditions under which the operator $D_{\mathrm{\Phi }}$ is self-adjoint and normal. Finally, we describe the spectrum of $D_{\mathrm{\Phi }}$ when the symbol $\mathrm{\Phi }(s)=c_1+c_2{2}^{-s}$.