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

Abstract

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

24 pp