Common reducing subspaces and decompositions of contractions

A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where $\mathbb E=\{(x_1,x_2,x_3)\in \mathbb C^3:\, 1-x_1z-x_2w+x_3zw \neq 0 \quad \text{ whenever } \; |z|\leq 1, \; \; |w|\leq 1 \} \subset \mathbb C^3,$ is a polynomially convex domain which is naturally associated with the $μ$-synthesis problem. By applications of the theory of $\mathbb E$-contractions, we obtain several results on decompositions of contractions.

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