An example of a domain of parreau-widom type

Abstract

Several theorems on the Hardy classes on the open unit disk have been extended to the Hardy classes on Riemann surfaces of Parreau–Widom type. However, some of the theorems, for instance, Beurling's invariant subspace theorem ([3,4, 7,9]) and the F. and M. Riesz theorem ([6]), are generalized under an additional condition. In this note we show that this additional condition cannot be dropped. Namely, we give an example of a plane domain which is of Parreau–Widom type and yet does not allow these theorems.