For \(0<r<1\) , let us consider the following annulus: \(\begin{aligned} {\mathbb {A}}_r= \{ z\in {\mathbb {C}}\, : \, r<|z|<1 \}. \end{aligned}\) A Hilbert space operator T for which \(\overline{{\mathbb {A}}}_r\) is a spectral set is called an \({\mathbb {A}}_r\) -contraction. Also, a normal operator U whose spectrum lies on the boundary \(\partial {\mathbb {A}}_r\) of \({\mathbb {A}}_r\) is called an \({\mathbb {A}}_r\) -unitary. We prove that any m number of commuting normal \({\mathbb {A}}_r\) -contractions \(N_1, \dots , N_m\) can be simultaneously dilated to commuting \({\mathbb {A}}_r\) -unitaries \(U_1, \dots , U_m\) . To construct such a dilation, we solve a Dirichlet problem for the polyannulus \({\mathbb {A}}_r^m\) . Also, we show that any finitely many doubly commuting subnormal \({\mathbb {A}}_r\) -contractions simultaneously dilate to commuting \({\mathbb {A}}_r\) -unitaries. Finally, we show that such a simultaneous \({\mathbb {A}}_r\) -unitary dilation holds for any finite number of doubly commuting \(2 \times 2\) scalar \({\mathbb {A}}_r\) -contractions.
