Jan-Fredrik Olsen: Hilbert spaces of Dirichlet series

Abstract: A Dirichlet series is a function of the type ∑ a_n n^(-s) where the summation is over n∈ℕ. A natural space of such functions, first considered by H. Hedenmalm, P. Lindqvist and K. Seip in 1997, is the Dirichlet-Hardy space

ℋ² = { ∑ a_n n^(-s) : (a_n) ∈ ℓ²}.

By the Cauchy- Schwarz inequality, the functions in this space are analytic for Re s > ½. A basic feature of this space is that the reproducing kernels are translates of the Riemann zeta
function. Another indication that the space ℋ² is interesting follows from the observation by H. Bohr in 1913 that there is a natural correspondence between Dirichlet series and power
series in infinitely many variables. This gives a natural identification of ℋ² with the Hardy space on the infinite- dimensional torus, H²(T^∞).

A result of importance, previously known by analytic number theorists and rediscovered by Hedenmalm, Lindqvist and Seip, says that functions in the Dirichlet-Hardy space are locally in L² on the abscissa Re s = ½. Later, it was shown by J.-F. Olsen and K. Seip that the Dirichlet-Hardy space and the classical Hardy space of the half- plane Re s > ½ have the same bounded interpolating sequences.

We discuss these two results, and their counter- parts for other Hilbert spaces of Dirichlet series, or equivalently, spaces of power series in infinitely many variables. Examples include the Bergman and Dirichlet spaces on the infinite- dimensonal torus, as well as the Drury-Arveson space on the infinite- dimensional unit ball.