Θ_n-contractions admit a canonical decomposition into a Θ_n-unitary plus a completely non-unitary part, and a Θ_n-isometric dilation exists exactly when a system of fundamental-operator equations has solutions satisfying four algebraic conditions.
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On the Dilation Theory and Canonical Decomposition of $\mathbf{\Theta}_n$-Contractions
Θ_n-contractions admit a canonical decomposition into a Θ_n-unitary plus a completely non-unitary part, and a Θ_n-isometric dilation exists exactly when a system of fundamental-operator equations has solutions satisfying four algebraic conditions.