A note on spectra of weighted composition operators on weighted Banach spaces of holomorphic functions
The author shows that, if \(v\) is a radial weight in the unit disc, nonincreasing in \(| z| \) and \(\lim_{| z| \to \infty}v(z)=0\), \(\phi:D\to D\) is an analytic self map which is not an automorphism and has a fixed point \(a\in D\), and \(\psi\) is analytic in \(D\) satisfying that \(C_{\phi,\psi}(f)=\psi (f\circ \phi)\) is bounded on \(H^\infty_v\), then the spectrum of the weighted composition operator \(C_{\phi,\psi}\) is given by the ball centered at the origin and radius the essential spectral radius together with the sequence \(\{\psi(a) \phi'(a)^n\}\). The proof follows the ideas from the previous works in the cases \(\psi(z)=1\) by \textit{R.\,Aron} and \textit{M.\,Lindström} for \(v(z)= (1-| z| ^2)^p\) [Isr.\ J.\ Math.\ 141, 263--276 (2004; Zbl 1074.47010)] and by \textit{J.\,Bonet, P.\,Galindo} and \textit{M.\,Lindström} for general weights [J.~Math.\ Anal.\ Appl.\ 340, No.\,2, 884--891 (2008; Zbl 1138.47017)].
spectra of weighted composition operators
weighted Banach spaces of analytic functions
scientific article; zbMATH DE number 5374767