Abstract
A tuple of commuting operators $$(S_1,\dots ,S_{n-1},P)$$ for which the closed symmetrized polydisc $$\Gamma _n$$ is a spectral set is called a $$\Gamma _n$$ -contraction. We show that every $$\Gamma _n$$ -contraction admits a decomposition into a $$\Gamma _n$$ -unitary and a completely non-unitary $$\Gamma _n$$ -contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $$\Gamma _n$$ and $$\Gamma _n$$ -contractions.
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