Leírás
We introduce a mean counting function for Dirichlet series, measuring a weighted density of their zeros in a half-plane. The existence of the mean is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. The mean counting function plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function plays in the classical theory. For example, the mean counting function yields a simple characterization of all (polydisc-)inner Dirichlet series, and the analogue of Frostman's theorem holds.
We then apply the mean counting function to resolve the problem of describing all compact composition operators with Dirichlet series symbols on the Hilbert-Hardy space of Dirichlet series, which has been a central problem of the theory since the bounded composition operators were described by J. Gordon and H. Hedenmalm in 1999. A composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary, in analogy with Shapiro's classical characterization of compact composition operators on the usual Hardy space.
Based on joint work with Ole Fredrik Brevig.
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