Dilation, functional model and a complete unitary invariant for C.₀Γ_n-contractions

A commuting tuple of operators (S1,...,Sn-1,P), defined on a Hilbert space H, for which the closed symmetrized polydisc Γn = {(Σ1≤i≤n zi, Σ1≤i<j≤n zizj,...,Π=1nzi): |zi|≤ 1,i = 1,...,n} is a spectral set, is called a Γn-contraction. A Γn-contraction is said to be pure or C.0 if P is C.0, that is, if P∗n → 0 strongly as n →∞. We show that for any Γn-contraction (S1,...,Sn-1,P), there is a unique operator tuple (A1,...,An-1) that satisfies the operator identities Si - Sn-i∗P = D PAiDP,i = 1,...,n - 1. This unique tuple is called the fundamental operator tuple or FO-tuple of (S1,...,Sn-1,P). With the help of the FO-tuple, we construct an operator model for a C.0 Γn-contraction and show that there exist n - 1 operators C1,...,Cn-1 such that each Si can be represented as Si = Ci + PCn-i∗. We find an explicit minimal dilation for a class of C.0 Γn-contractions whose FO-tuples satisfy a certain condition. Also, we establish that the FO-tuple of (S1∗,...,S n-1∗,P∗) together with the characteristic function of P constitutes a complete unitary invariant for the C.0 Γn-contractions. The entire program is an analog of the Sz.-Nagy-Foias theory for C.0 contractions.