Decomposition for a composition operator (Q404496)

Let \(\varphi\) denote an elliptic automorphism of the unit disk of the form \(\varphi(z)=\varphi_a(z)\circ\rho_w(z)\circ\varphi_a(z)\), where \(\varphi_a(z)=(a-z)/(1-\bar az)\) with \(|a|<1\), \(w\) is the \(k\)-th root of unity with \(k\geq2\), and \(\rho_w(z)=wz\). The paper shows that the composition operator \(C_\varphi\) on the Hardy space \(H^2\) admits the decomposition \(C_\varphi=\sum_{s=0}^{k-1}w^sP_s\), where each \(P_s\) is an orthogonal projection.