Composition-Differentiation Operator On Hardy-Hilbert Space of Dirichlet Series
Vasudevarao Allu, Dipon Kumar Mondal
TL;DR
This work studies the composition-differentiation operator on the Hardy-Hilbert space of Dirichlet series $H^2$, establishing a boundary-decay criterion for compactness via the mean counting function $M_Φ$ and deriving boundedness criteria for zero-characteristic symbols. It specializes to the affine symbol $Φ(s)= c_1 + c_2 2^{-s}$ to obtain explicit norm bounds, approximation-number estimates, and spectral results, and it provides a complete characterization of when $D_Φ$ is self-adjoint or normal. The spectrum is shown to be {0} for the affine symbol under a growth condition on $Re(c_1)$, with the operator being quasinilpotent and not normal in this regime. Overall, the paper extends compactness, norm, and spectral analysis from classical Hardy spaces to the Hardy-Hilbert space of Dirichlet series, delivering precise quantitative bounds for $D_Φ$.
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 \( Φ(s) = c_1 + c_2 2^{-s} \).