Joint reducing subspaces and orthogonal decompositions of operators in an annulus

A commuting tuple of Hilbert space operators $(T_1, \dotsc, T_n)$ is said to be 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 their structures. We find Wold type decompositions for any number of commuting and doubly commuting $\mathbb A_r$-isometries. Then we generalize these results to any family of commuting and doubly commuting $\mathbb A_r$-contractions.