Composition-differentiation operators on Hardy-Hilbert space of Dirichlet series

Abstract

In this paper, we establish a compactness criterion for the composition-differentiation operator $D_{\Phi }$ on the Hardy space ${\mathcal{H}}^2$ of Dirichlet series in terms of a boundary decay condition for its mean counting function. Via a comparison-type principle, we show that this boundary decay is equivalent to a corresponding decay condition for the Green's function, which we further analyze through harmonic measure. We provide explicit mapping properties of the symbol $\Phi$ that generate a bounded composition-differentiation operator $D_{\Phi }$ and obtain precise norm estimates for $D_{\Phi }$ when $\Phi$ is an affine symbol with a single-prime in the class ${\mathcal{G}}_0$. Furthermore, we establish explicit upper and lower bounds for the approximation numbers of $D_{\Phi }$ on ${\mathcal{H}}^2$ motivated by the work of Queffélec and Seip [J. Funct. Anal., 2015]. Finally, we investigate spectral and operator-theoretic properties of $D_{\Phi }$ for symbols in ${\mathcal{G}}_0$.

The paper is now thoroughly revised. In particular, a few new techniques are used in Sections 3 and 4 compared to the previous version, and Section 3 now incorporates a key new result (Theorem 3.2). Furthermore, the proof of Theorem 2.4 has been revised to be more geometric, and Theorem 2.3 offers two applications