Hardy Spaces of Dirichlet Series and Their Composition Operators
- Cite this article as:
- Bayart, F. Mh Math (2002) 136: 203. doi:10.1007/s00605-002-0470-7
In , 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 () about the hypercontractivity of this semigroup on the spaces \(\). Finally, following , we determine the composition operators on \(\), and we compare some properties of such an operator and of its symbol.