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 byTf (z) = m(z)f (beta z), (z E D)where m E Hol(D), beta E C, |beta| = 1. If beta n * 1 for all n E N we show that sigma W(T) is a disc if m(z0) = 0 for some z0 E D, and it is the circle {lambda E C : |lambda| = |m(0)|} if m(z) * 0 for all z E D. We find examples of m E A(D) (the disc algebra) such that lambda Id -T is invertible in Hol(D) (the Fre ' chet space of all holomorphic functions on D), but (lambda Id -T )-1A(D) ct A(D). Inspired by Bonet [2] we show that {beta n : n E N} c sigma (T) * T when the weight is m equivalent to 1 and beta a Diophantine number. This shows that the spectrum is not closed in general.