Spectra of Some Weighted Composition Operators on $H^2$

We completely characterize the spectrum of a weighted composition operator $W_{\psi, \varphi}$ on $H^{2}(\mathbb{D})$ when $\varphi$ has Denjoy-Wolff point $a$ with $0<|\varphi '(a)|< 1$, the iterates, $\varphi_{n}$, converge uniformly to $a$, and $\psi$ is in $H^{\infty}(\mathbb{D})$ and continuous at $a$. We also give bounds and some computations when $|a|=1$ and $\varphi '(a)=1$ and, in addition, show that these symbols include all linear fractional $\varphi$ that are hyperbolic and parabolic non-automorphisms. Finally, we use these results to eliminate possible weights $\psi$ so that $W_{\psi, \varphi}$ is seminormal.