Eigenvalues of Adjoints of Certain Composition Operators and Weighted Composition Operators

We look at the point-spectrum of the adjoints of certain composition operators on the Hardy–Hilbert space, for which the composition map has two distinguished fixed points: one inside the unit disk and one on the unit circle. In particular, we show that the point-spectrum of such operators contains a disk centered at the origin, and each eigenvalue in that disk has infinite multiplicity. We also identify for every such operator a subspace of the Hardy–Hilbert space which is invariant for the operator and on which it acts like a weighted shift. Finally, we generalize these results to weighted composition operators.