Spectral properties of weighted composition operators on $\Hol(\D)$ induced by rotations

Abstract

In this article we study the spectrum $\ddot{I}ƒ(T)$ and Waelbroeck spectrum $\ddot{I}{ƒ}_W(T)$ of a weighted composition operator $T$ induced by a rotation on $\Hol(\D)$ and given by $$Tf(z)=m(z)f(βz) \ \ \ (z\in \D)$$ where $m\in \Hol(\D)$, $β\in \C$, $|{\hat{I}}^2|=1$. If $β^n\neq 1$ for all $n\in \mathbb{N}$ we show that $\ddot{I}{ƒ}_W(T)$ is a disc if $m(z_0)=0$ for some $z_0\in \D$ and it is the circle $\{λ\in \C : |λ|=|m(0)|\}$ if $m(z)\ne 0$ for all $z\in \D$. We find examples of $m\in A(\D)$ (the disc algebra) such that $λ\Id-T$ is invertible in $\Hol(\D)$ (the Fréchet space of all holomorphic functions on $\D$), but $(λ\Id-T)^{-1}A(\D)\not\subset A(\D)$. Inspired by Bonet \cite{Bonet} we show that $\{β^n : n\in \N\}\subset σ(T)\neq \T$ when the weight is $m\equiv 1$ and ${\hat{I}}^2$ a diophantine number. This shows that the spectrum is not closed in general.