Spectrum of a compact weighted composition operator

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