On the Beurling-Lax theorem for domains with one hole

摘要

We consider pure subnormal operators T ofthe type studied in Carlsson, 2011, with the additional requirementthat σ(T) has one hole. If ind(T-λ0)=-n for someλ0 and n∈N, we show that the operator can bedecomposed as T=⊕k=1n Tk, where each Tk satisfiesind(T-λ0)=-1, thus extending the classical Beurling-Laxtheorem (in which σ(T) is the unit disc). We also provide aset of unitary invariants that completely characterize T and studythe model spaces for the simpler operators Tk.