We consider pure subnormal operators T of the type studied in Carlsson, 2011, with the additional requirement that σ(T) has one hole. If ind(T-λ₀)=-n for some λ₀ and n∈N, we show that the operator can be decomposed as T=⊕_k=1ⁿ T_k, where each T_k satisfies ind(T-λ₀)=-1, thus extending the classical Beurling-Lax theorem (in which σ(T) is the unit disc). We also provide a set of unitary invariants that completely characterize T and study the model spaces for the simpler operators T_k.

This research was supported by the Swedish research council (2008-23883-61232-34) and the Swedish Foundation for International Cooperation in Research and Higher Education (YR2010-7033).