Composition operators on weighted spaces of holomorphic functions on the upper half plane - Lusky, Wolfgang (MATHEMATICA SCANDINAVICA, 2018)
We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(Im w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{φ }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $φ \colon \mathbb G → \mathbb G$ where the composition operator $C_{φ } $ is a bounded operator $Hv(\mathbb G) → Hv(\mathbb G)$ by a simple property which depends only on $φ $ but not on $v$. Moreover we show that there are no compact composition operators $C_{φ }$ on $Hv(\mathbb G)$.