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

TL;DR

This paper investigates the structure and approximation properties of composition operators on the Hilbert space of Dirichlet series, providing orthogonal decompositions, asymptotic estimates, and criteria for compactness based on the symbol's prime support.

Contribution

It introduces an orthogonal decomposition for composition operators with symbols supported on prime subsets and characterizes their compactness, advancing understanding of their spectral and approximation properties.

Findings

Orthogonal decomposition of composition operators based on prime support.

Asymptotic estimates of approximation numbers under sparseness conditions.

Characterization of compactness via Nevanlinna counting function for single-prime supported symbols.

Abstract

Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $\mathscr{C}_\varphi$ on $\mathscr{H}^2$ which are generated by symbols of the form $\varphi(s) = c_0s + \sum_{n\geq1} c_n n^{-s}$, in the case that $c_0 \geq 1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet series of $\varphi$, then the operator $\mathscr{C}_\varphi$ admits an associated orthogonal decomposition. Under sparseness assumptions on $\mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $\mathscr{C}_\varphi$. Furthermore, in the case that $\varphi$ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of $\mathscr{C}_\varphi$ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.