Composition operators on weighted Hilbert spaces of Dirichlet series

Abstract

We study composition operators of characteristic zero on weighted Hilbert spaces of Dirichlet series. For this purpose we demonstrate the existence of weighted mean counting functions associated with the Dirichlet series symbol, and provide a corresponding change of variables formula for the composition operator. This leads to natural necessary conditions for the boundedness and compactness. For Bergman-type spaces, we are able to show that the compactness condition is also sufficient, by employing a Schwarz-type lemma for Dirichlet series.

Composition operators and generalized primes

“…This and ( 20) imply (19). Again, by the Kellogg-Warschawski theorem there exists r > 0 such that (18) follows by the Koebe quarter theorem working as above.…”

“…Our argument will rely on a technique which has been developed in [9,19] and allows us to transfer our notions in the disk setting.…”

“…Despite its technical and maybe serendipitous look, the idea behind Lemma 4.5 may be useful. See, for example the interchange of limits problem [9, Problem 1] and the partial solution of it [19,Theorem 4.9].…”

Counting functions for Dirichlet series and compactness of composition operators

“…The converse is true if 𝜑 is supported on a finite set of prime numbers. The case 𝜑 ∈  0 is thoroughly studied in [11], using a counting function in the spirit of [7].…”