Spectrum of a compact weighted composition operator

For $\psi$ analytic in the open unit disk and $\varphi$ an analytic map from the unit disk into itself, the weighted composition operator $C_{\psi,\varphi}$ is the operator on the weighted Hardy space $H^{2}(\beta)$ given by $(C_{\psi,\varphi}f)(z)=\psi(z)f(\varphi(z)).$ This paper discusses the spectrum of $C_{\psi,\varphi}$ when it is compact on a certain class of weighted Hardy spaces and when the composition map $\varphi$ has a fixed point inside the open unit disk.