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

摘要

Let H-2 denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators l(phi) on H-2 which are generated by symbols of the form phi(s) = c(0)s + Sigma(n >= 1) c(n)n(-8), in the case that c(0) >= 1. If only a subset P of prime numbers features in the Dirichlet series of l(phi), then the operator l(phi) admits an associated orthogonal decomposition. Under sparseness assumptions on P we use this to asymptotically estimate the approximation numbers of l(phi). Furthermore, in the case that phi is supported on a single prime number, we affirmatively settle the problem of describing the compactness of if l(phi) in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps. (c) 2021 The Author(s). Published by Elsevier Inc.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).