Spectra of some weighted composition operators on H²

We completely characterize the spectrum of a weighted composition operator W Ψ,ℓ on H²(D) when ℓ has Denjoy-Wolff point a with 0 < |ℓ'′(α)| < 1, the iterates, ℓ_n, converge uniformly to a, and is in H∞ (the space of bounded analytic functions on D) and continuous at a. We also give bounds and some computations when |α| = 1 and ℓ'′(α) = 1 and, in addition, show that these symbols include all linear fractional ℓ that are hyperbolic and parabolic nonautomorphisms. Finally, we use these results to eliminate possible weights Ψ so that W Ψ,ℓ is seminormal.