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

Abstract

We completely characterize the spectrum of a weighted composition operator $W_{\psi ,\phi }$ on $H^2(\mathbb{D})$ when $\phi$ has Denjoy-Wolff point $a$ with $0<|{\phi }^{\prime }(a)|<1$, the iterates, ${\phi }_n$, 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 ${\phi }^{\prime }(a)=1$ and, in addition, show that these symbols include all linear fractional $\phi$ that are hyperbolic and parabolic non-automorphisms. Finally, we use these results to eliminate possible weights $\psi$ so that $W_{\psi ,\phi }$ is seminormal.

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• Quasinormal and hyponormal weighted composition operators on $H^2$ and $A_{\alpha }^2$ with linear fractional compositional symbol