Composition operators and generalized primes

Abstract

We study composition operators on the Hardy space H 2 \mathcal {H}^2 of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on H 2 \mathcal {H}^2 . To do that we extend our notions to a Hardy space H Λ 2 \mathcal {H}_{\Lambda }^2 of generalized Dirichlet series, induced in a natural way by a sequence of Beurling’s primes.