Invertible weighted composition operators (Q5891301)

The author considers composition and weighted composition operators acting on sets of analytic functions rather than vector spaces or normed spaces. As usual, if \(X\) is a set of analytic functions in the open unit disc, an analytic function \(\phi:\mathbb D\to \mathbb D\) is said to induce a composition operator in \(X\) provided that \(C_\phi (f)=f\circ \phi \in X\) whenever \(f\in X\) and, similarly, a couple \(\phi, \psi\) is said to induce a weighted composition operator in \(X\) if \(W_{\psi,\phi} f = \psi C_\phi(f)\in X\) whenever \(f\in X\). The author's main results establish some sufficient conditions on the set \(X\) which allow to conclude that the invertibility of \(C_\phi\) or \(W_{\psi, \phi}\) in \(X\) guarantees that \(\phi\) is an automorphism. He applies this rather abstract formulation to get results whenever \(X\) are Hardy or weighted Hardy spaces \(H^2(\beta)\), recovering some known results (see [\textit{G. Gunatillake}, J. Funct. Anal. 261, No. 3, 831--860 (2011; Zbl 1218.47037)]) and also new results in the case \(X=\{f\in \mathcal H(\mathbb D): f'\in H^p\}\), getting in most cases complete characterizations of the invertibility of composition and weighted composition on these spaces.