A Nagy–Foias Program for a C.N.U. Γn-Contraction

A tuple of commuting Hilbert space operators (S₁,…,Sn-1,P) having the closed symmetrized polydisc (Formula presented.) as a spectral set is called a Γ_n-contraction. From the literature we have that a point (s₁,…,sn-1,p) in Γ_n can be represented as s_i=c_i+pcn-i for some (c₁,…,cn-1)∈Γn-1. We construct a minimal Γ_n-isometric dilation for a particular class of c.n.u. Γ_n-contractions (S₁,…,Sn-1,P) and obtain a functional model for them. With the help of this model we express each S_i as S_i=C_i+PCn-i, which is an operator theoretic analogue of the scalar result. We also produce an abstract model for a different class of c.n.u. Γ_n-contractions satisfying S_i^∗P=PS_i^∗ for each i. By exhibiting a counter example we show that such abstract model may not exist if we drop the hypothesis that S_i^∗P=PS_i^∗. We apply this abstract model to achieve a complete unitary invariant for such c.n.u. Γ_n-contractions. Additionally, we present different necessary conditions for dilation and a sufficient condition under which a commuting tuple (S₁,…,Sn-1,P) becomes a Γ_n-contraction. The entire program goes parallel to the operator theoretic program developed by Sz.-Nagy and Foias for a c.n.u. contraction.