On decomposition of operators having ${\mathrm{\Gamma }}_3$ as a spectral set

The symmetrized polydisc of dimension three is the set $${\mathrm{\Gamma }}_3=\{(z_1+z_2+z_3,z_1{z}_2+z_2{z}_3+z_3{z}_1,z_1{z}_2{z}_3):\mathrm{\mid }{z}_i\mathrm{\mid }\le 1,i=1,2,3\}\subseteq {\mathbb{C}}^3\mathrm{.}$$ A triple of commuting operators for which ${\mathrm{\Gamma }}_3$ is a spectral set is called a ${\mathrm{\Gamma }}_3$-contraction. We show that every ${\mathrm{\Gamma }}_3$-contraction admits a decomposition into a ${\mathrm{\Gamma }}_3$-unitary and a completely non-unitary ${\mathrm{\Gamma }}_3$-contraction. This decomposition parallels the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set ${\mathrm{\Gamma }}_3$ and ${\mathrm{\Gamma }}_3$-contractions.