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

Abstract

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

Derek Thompson is the primary contact person the paper