On the Beurling–Lax theorem for domains with one hole

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=1Tk, where each Tk 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 Tk.