Orthogonal decomposition of composition operators on the $H^2$ space of Dirichlet series

Abstract

Let ${\mathcal{H}}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators ${\mathcal{C}}_{\phi }$ on ${\mathcal{H}}^2$ which are generated by symbols of the form $\phi (s)=c_{0}s+{\sum }_{n\ge 1}{c}_n{n}^{-s}$, in the case that $c_0\ge 1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet series of $\phi$, then the operator ${\mathcal{C}}_{\phi }$ admits an associated orthogonal decomposition. Under sparseness assumptions on $\mathbb{P}$ we use this to asymptotically estimate the approximation numbers of ${\mathcal{C}}_{\phi }$. Furthermore, in the case that $\phi$ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of ${\mathcal{C}}_{\phi }$ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.

arXiv commentThis paper has been accepted for publication in Journal of Functional Analysis

References in corpus (5)

• Approximation numbers of composition operators on the $H^2$ space of Dirichlet series
• A mean counting function for Dirichlet series and compact composition operators
• Decay rates for approximation numbers of composition operators
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Cited by in corpus (1)

• Counting functions for Dirichlet series and compactness of composition operators