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

Abstract

Let denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators on which are generated by symbols of the form , in the case that . If only a subset of prime numbers features in the Dirichlet series of φ, then the operator admits an associated orthogonal decomposition. Under sparseness assumptions on we use this to asymptotically estimate the approximation numbers of . Furthermore, in the case that φ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.