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

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