Actions of the Möbius group on analytic functions
Volume 260 / 2021
Studia Mathematica 260 (2021), 207-228
MSC: 30H20, 30H25, 30H30, 30H35, 30B10, 46E15.
DOI: 10.4064/sm200924-21-12
Published online: 4 March 2021
Abstract
In his PhD dissertation (1996), Ruhan Zhao introduced a new notion of weighted actions by the Möbius group on analytic functions on the unit disk indexed by a positive parameter $\alpha $ and proved that the so-called $\alpha $-Bloch space is maximal among all $\alpha $-Möbius invariant function spaces. In this paper we continue the study of $\alpha $-Möbius invariant function spaces. In particular, we identify the minimal non-trivial $\alpha $-Möbius invariant function space and prove the existence and uniqueness of a non-trivial $\alpha $-Möbius invariant semi-Hilbert space of analytic functions on the unit disk, thus answering two questions left open by Zhao.
