Dilation of normal operators associated with an annulus

For $0<r<1$, let us consider the following annulus: $$\mathbb A_r= \{ z\in \mathbb C\, : \, r<|z|<1 \}.$$ A Hilbert space operator $T$ for which $\overline{\mathbb A}_r$ is a spectral set is called an $\mathbb A_r$-\textit{contraction}. Also, a normal operator $U$ whose spectrum lies on the boundary $\partial \mathbb A_r$ of $\mathbb A_r$ is called an $\mathbb A_r$-\textit{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.