Joint reducing subspaces and orthogonal decompositions of operators in an annulus

A commuting tuple of Hilbert space operators $(T_1,\dots ,T_n)$ is said to be an ${\mathbb{A}}_r^n\mathit{\text{-contraction}}$ if the closure of the polyannulus $${\mathbb{A}}_r^n=\{(z_1,\dots ,z_n):r<\mathrm{\mid }{z}_i\mathrm{\mid }<1,1\le i\le n\}\subseteq {\mathbb{C}}^n(0<r<1)$$ is a spectral set for $(T_1,\dots ,T_n)$. 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.