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

We study the spectrum σ(T) of a weighted composition operator T induced by a weight m∈Hol(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 D, we show that σ(T) is a bounded discrete set such that σ(T)∖{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 C∖{0} whenever m≠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 C∖{0}.