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A commuting tuple of Hilbert space operators $(T_1, \dotsc, T_n)$ is said tobe an \textit{$\mathbb{A}_r^n$-contraction} if the closure of the polyannulus  \[  \mathbb A_r^n=\left\{(z_1, \dotsc, z_n) \ : \ r<|z_i|<1, \ 1 \leq i \leq n\right\} \subseteq \mathbb{C}^n \qquad \quad (0<r<1)  \]  is a spectral set for $(T_1, \dotsc, T_n)$. We find characterizations for the$\mathbb A_r^n$-unitaries and $\mathbb A_r^n$-isometries and decipher theirstructures. We find Wold type decompositions for any number of commuting anddoubly commuting $\mathbb A_r$-isometries. Then we generalize these results toany family of commuting and doubly commuting $\mathbb A_r$-contractions.

Joint reducing subspaces and orthogonal decompositions of operators in an annulus