Essential Spectra of Weighted Composition Operators Induces by Elliptic Automorphisms

The spectrum of a weighted composition operator $C_{\psi ,\phi }$ who is induced by an automorphism has been investigated for over fifty years. However, many results are got only under the condition that the weight function $\psi$ is continuous up to the boundary. In this paper we study the spectra and essential spectra of $C_{\psi ,\phi }$ on weighted Bergman spaces when $\phi$ is an elliptic automorphism, without the assumption that $\psi$ is continuous up to the boundary.