A Hilbert space operator T is said to be an \(\mathbb {A}_r\) -contraction if the closure of the annulus \( \mathbb {A}_r=\{z \in \mathbb {C} \ : \ r<|z|<1\} \qquad (0<r<1) \) is a spectral set for T. An \(\mathbb {A}_r\) -unitary is a normal operator with spectrum inside the boundary \(\partial \mathbb {A}_r\) . An \(\mathbb {A}_r\) -isometry is a subnormal \(\mathbb {A}_r\) -contraction whose minimal normal extension is an \(\mathbb {A}_r\) -unitary. A pure \(\mathbb {A}_r\) -isometry is an \(\mathbb {A}_r\) -isometry that has no \(\mathbb {A}_r\) -unitary part. Agler proved the success of rational dilation on \(\overline{\mathbb {A}}_r\) , i.e., every \(\mathbb {A}_r\) -contraction admits a normal boundary dilation. For the class of invertible operators \(C_{1, r}=\{T: \Vert T\Vert , \Vert rT^{-1}\Vert \le 1\}\) and \( C_\alpha =\{T: \ T~{\text {is invertible and}} \ \alpha (T^*, T)=-T^{*2}T^2+(1+r^2)T^*T-r^2I \ge 0\}. \) Bello and Yakubovich proved that \(\{T: \ T \ {\text {is an}}~ \mathbb {A}_r-{\text {contraction}}\} \subsetneq C_\alpha \subsetneq C_{1,r}\) and provided a model theorem for \(C_\alpha \) class of operators. Capitalizing their model for operators in \(C_\alpha \) , we obtain a model theorem for \(\mathbb {A}_r\) -contractions T satisfying \(T^n \rightarrow 0\) and \((rT^{-1})^n \rightarrow 0\) strongly as \(n \rightarrow \infty \) . Models for \(\mathbb {A}_r\) -isometries and pure \(\mathbb {A}_r\) -isometries are provided. Furthermore, canonical decomposition and Levan type decomposition are found for operators in \(C_\alpha \) class.
