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

TL;DR

This paper investigates the properties of the composition-differentiation operator on Hardy-Hilbert spaces of Dirichlet series, providing criteria for compactness, boundedness, and spectral characteristics based on the symbol function.

Contribution

It establishes new criteria for compactness and boundedness of the operator, and explicitly characterizes its spectrum for specific symbols.

Findings

Established a compactness criterion based on decay of mean counting function.

Provided a sufficient condition for boundedness when the symbol has zero characteristic.

Derived explicit spectrum for the operator with specific symbol  

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