Hardy Spaces Of Dirichlet Series Research Articles

On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate

A range of Hardy-like spaces of ordinary Dirichlet series, called the Dirichlet–Hardy spaces ℋp, p ≧ 1, have been the focus of increasing interest among researchers following a paper of Hedenmalm, Lindqvist and Seip [16]. The Dirichlet series in these spaces converge on a certain half-plane, where one may also define the classical Hardy spaces Hp. In this paper, we compare the boundary behaviour of elements in ℋp and Hp. Moreover, Carleson measures for the spaces ℋp are studied. Our main result shows that for certain cases the following statement holds true. Given an interval on the boundary of the half-plane of definition and a function in the classical Hardy space, it is possible to find a function in the corresponding Dirichlet–Hardy space such that their difference has an analytic continuation across this interval.

On the shift semigroup on the Hardy space of Dirichlet series

We develop a Wold decomposition for the shift semigroup on the Hardy space \( \mathcal{H}^2 \) of square summable Dirichlet series convergent in the half-plane \( \Re (s) > 1/2 \). As an application we have that a shift invariant subspace of \( \mathcal{H}^2 \) is unitarily equivalent to \( \mathcal{H}^2 \) if and only if it has the form \( \phi \mathcal{H}^2 \) for some \( \mathcal{H}^2 \)-inner function φ.

Hardy Spaces of Dirichlet Series and Their Composition Operators

In [9], Hedenmalm, Lindqvist and Seip introduce the Hilbert space of Dirichlet series with square summable coefficients , and begin its study, with modern functional and harmonic analysis tools. The space is an analogue for Dirichlet series of the space for Fourier series. We continue their study by introducing , an analogue to the spaces . Thanks to Bohr’s vision of Dirichlet series, we identify with the Hardy space of the infinite polydisk . Next, we study a variant of the Poisson semigroup for Dirichlet series. We give a result similar to the one of Weissler ([25]) about the hypercontractivity of this semigroup on the spaces . Finally, following [8], we determine the composition operators on , and we compare some properties of such an operator and of its symbol.