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

2025/02/27 by Allu, Vasudevarao, Mondal, Dipon Kumar
#30B50 #30H10 #47B33 #47B38 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2502.19939

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 \( Φ(s) = c₁ + c₂ 2^-s \). 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 \( Φ(s) = c₁ + c₂ 2^-s \).

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• Products of composition and differentiation operators on the Hardy space
• Composition operators on the algebra of Dirichlet series
• Weighted Composition--Differentiation Operator on the Hardy and Weighted Bergman Spaces
• Topological structure of the space of composition operators on the Hardy space of Dirichlet series