A commuting tuple of operators (S-1, ..., Sn-1, P), defined on a Hilbert space H, for which the closed symmetrized polydisc Gamma(n) = {(Sigma(1 0 strongly as n -> infinity. We show that for any Gamma(n)-contraction (S-1, ..., Sn-1, P), there is a unique operator tuple (A(1), ..., A(n-1)) that satisfies the operator identities S-i - Sn-i*P = D(P)A(i)D(P), i = 1, ..., n - 1. This unique tuple is called the fundamental operator tuple or F-O-tuple of (S-1, ..., Sn-1, P). With the help of the F-O-tuple, we construct an operator model for a C.0 Gamma(n)-contraction and show that there exist n - 1 operators C-1, ..., Cn-1 such that each S-i can be represented as S-i = C-i + PCn-i*. We find an explicit minimal dilation for a class of C.(0) Gamma(n)-contractions whose F-O-tuples satisfy a certain condition. Also, we establish that the F-O-tuple of (S-1*, ..., Sn-1*, P*) together with the characteristic function of P constitutes a complete unitary invariant for the C.(0) Gamma(n)-contractions. The entire program is an analog of the Sz.-Nagy-Foias theory for C.(0) contractions.
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