Actions of the Möbius group on analytic functions

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 α and proved that the so-called α-Bloch space is maximal among all α-Möbius invariant function spaces. In this paper we continue the study of α-Möbius invariant function spaces. In particular, we identify the minimal non-trivial α-Möbius invariant function space and prove the existence and uniqueness of a non-trivial α-Möbius invariant semi-Hilbert space of analytic functions on the unit disk, thus answering two questions left open by Zhao.