The Hardy-Dirichlet space \(\mathcal{H}^p\) and its composition operators. (English) Zbl 1449.32001
Jaming, Philippe (ed.) et al., Harmonic analysis, function theory, operator theory, and their applications. Proceedings of the international conference in honour of Jean Esterle, Bordeaux, France, June 1–4, 2015. Editor list corrected in erratum. Bucharest: The Theta Foundation. Theta Ser. Adv. Math. 19, 221-250 (2017).
Summary: We present some recent results on composition operators acting on a Hardy space \(\mathcal{H}^p\) of a new type, formed by Dirichlet series. This study was initiated by Hedenmalm-Lindqvist-Seip for \(p = 2\) to answer a question of Beurling, and then it was continued for \(p\neq 2\) by Bayart. We get new results on the spectrum and the approximation numbers of such operators, especially when \(p = 1\). The proofs use interpolation sequences, Carleson measures and extensions of the Weyl inequalities to the Banach space setting, as well as the prime number theorem. Many interesting problems remain open.
For the entire collection see [Zbl 1404.42002].
For the entire collection see [Zbl 1404.42002].
MSC:
| 32A05 | Power series, series of functions of several complex variables |
| 43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |
