Common reducing subspaces and decompositions of contractions

Abstract

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 $\mu$-synthesis problem. By applications of the theory of $\mathbb{E}$-contractions, we obtain several results on decompositions of contractions.

24 pages, Forum Mathematicum, To appear