Denjoy–Wolff theory and spectral properties of weighted composition operators on Hol(D)

Abstract

We study the spectrum $\mathit{\sigma }(T)$ of a weighted composition operator T induced by a weight $m\in \mathrm{Hol}(\mathbb{D})$ and a holomorphic self-map φ on the unit disc, which is not an elliptic automorphism. If φ has a unique fixed point in $\mathbb{D}$, we show that $\mathit{\sigma }(T)$ is a bounded discrete set such that $\mathit{\sigma }(T)\setminus \{0\}$ is a set of eigenvalues with multiplicity one. If φ has a Denjoy–Wolff point α on the unit circle, we first prove that the point spectrum is $\mathbb{C}\setminus \{0\}$ whenever $m\ne 0$ is constant. Moreover, the multiplicity of each eigenvalue is infinite. Then we describe classes of m for which the point spectrum of T is either empty or equal to $\mathbb{C}\setminus \{0\}$.