Characterizations and models for the C₁r class and quantum annulus

For fixed 0 < r < 1, let Ar = {z ∈ C ∶ r < ∣z∣ < 1} be the annulus with boundary ∂Ar = T ∪ rT, where T is the unit circle in the complex plane C. An operator having Ar as a spectral set is called an Ar-contraction. Also, a normal operator with its spectrum lying in the boundary ∂Ar is called an Ar-unitary. The C1,r class was introduced by Bello and Yakubovich in the following way: C1,r = {T ∶ T is invertible and ∥T∥, ∥rT⁻¹∥ ≤ 1}. McCullough and Pascoe defined the quantum annulus QAr by QAr = {T ∶ T is invertible and ∥rT∥, ∥rT⁻¹∥ ≤ 1}. If Ar denotes the set of all Ar-contractions, then Ar C1,r QAr. We first find a model for an operator in C1,r and also characterize the operators in C1,r in several different ways. We prove that the classes C1,r and QAr are equivalent. Then, via this equivalence, we obtain analogous model and characterizations for an operator in QAr.