Essential spectrum and Fredholm index for certain composition operators (Q2795656)

For \(0<\alpha<1\), \(0<\sigma<1\), \(c_1\) so that \(\alpha<c_1<1\) and \(\gamma=\sigma\alpha\), let \(U\) be a subdomain of the open unit disc \(\mathbb D=\{z\in\mathbb C:| z|<1\}\) defined by \(\mathbb D\setminus\{0\}\setminus \bigcup_{n\geq 1}D_n\) where \(D_n=\{| z-\alpha^{n-1}c_1|\leq\gamma^n\}\). The author studies the spectrum and essential spectrum of the composition operator \(C_\phi(f)(z)=f(\phi(z))\) for \(f\in H^\infty(U)\) where \(\phi(z)=\alpha z\). The answers obtained are expressed in terms of geometric quantities related to \(\alpha\) and \(\sigma\).