Univalently induced, closed range, composition operators on the Bloch-type spaces (Q2888799)

For any positive \(\alpha\), the space \(B^\alpha\) consists of all analytic functions in the unit disk \(D\) such that NEWLINE\[NEWLINE\sup_{z\in D}|f'(z)|(1-|z|^2)^\alpha<\infty.NEWLINE\]NEWLINE The main result of the paper states that, if \(\alpha\not=1\) and \(\varphi\) is a univalent self-map of the unit disk, then the composition operator \(C_\varphi:B^\alpha\to B^\alpha\) has closed range if and only if \(\varphi\) is a Möbius map.NEWLINENEWLINENote that when \(\alpha=1\), \(B^\alpha\) is the classical Bloch space. In this case, there exist many non-Möbius univalent self-maps of the unit disk that induce composition operators with closed range on the Bloch space.