Spectra of weighted composition operators with automorphic symbols

Let φ be an automorphism of the open unit disc D. For such φ, we investigate the spectra of invertible weighted composition operators uCφ acting on a wide class of analytic function spaces; this class contains, for example, Hardy spaces Hp(D), weighted Bergman spaces Aαp(D), and weighted Banach spaces of H∞-type. We present new techniques for deducing the spectrum and for calculating the spectral radius of uCφ. We also characterize the Fredholmness of weighted composition operators on Hp(D) and Aαp(D).