Spectra of weighted composition operators on weighted Banach spaces of analytic functions (Q1885660)

Let \(v_p=(1-| z| ^2)^p \;(p>0)\) and denote \(H^{\infty}_v=\sup\{f\in H(D): \| f\| _{v_p}:=\sup_{z\in D}v_p(z)| f(z)| <\infty\}.\) Suppose that \(\phi: D\rightarrow D\), not an automorphism, has a fixed point \(a\in D\), \(\psi\in H(D)\) and the weighted composition operator \(C_{\psi,\phi}: H^{\infty}_{v_p}\rightarrow H^{\infty}_{v_p}\) is bounded. Then the spectra of \(C_{\psi,\phi}\) satisfy \[ \delta_{H^{\infty}_{v_p}}(C_{\psi,\phi})=\{\lambda\in{\mathbb C}: | \lambda| \leq r_{e,H^{\infty}_{v_p}} (C_{\psi,\phi})\}\cup\{\psi(a)\phi^{\prime}(a)^n\}^{\infty}_{n=0}, \] where \(r_{e,H^{\infty}_{v_p}}\) is the essential norm of \(C_{\psi,\phi}\). At the same time, this result can be applied to give the spectra of composition operators on Bloch type spaces which proves in the affirmative a conjecture given by \textit{B. MacCluer} and \textit{K. Saxe} [Isr. J. Math. 128, 325--354 (2002; Zbl 1024.47009)].