The fundamental operator tuples associated with the symmetrized polydisc.

A commuting tuple of operators (S1; ....; Sn-1; P), defined on a Hilbert space H, for which the closed symmetrized polydisc is a spectral set, is called a Γn-contraction. To every Γn-contraction, there is a unique operator tuple (A1;...;An-1), defined on Ran(I -P* P), such that Si - S*n-iP = DPAiDP ; DP = (I - P*P) 1 2 ; i = 1; ....; n - 1: This is called the fundamental operator tuple or FO-tuple associated with the Γn-contraction. The FO-tuple of a Γn-contraction completely deter- mines the structure of a Γn-contraction and provides operator model and complete unitary invariant for them. In this note, we analyze the FO-tuples and defind some intrinsic properties of them. Given a Γn-contraction (S1; ....; Sn-1; P) and n-1 operators A1; ....;An-1 defined on RanDP, we provide a necessary and suffcient condition under which (A1; ....;An-1) becomes the FO-tuple of (S1;.... ; Sn-1; P). Also for given tuples of operators (A1; ....;An-1) and (B1; ....;Bn-1), defined on a Hilbert space E, we find a necessary condition and a sufficient condition under which there exist a Hilbert space H and a Γn-contraction (S1;....; Sn-1; P) on H such that (A1;... ;An-1) becomes the FO-tuple of (S1;....; Sn-1; P) and (B1; ....;Bn-1) becomes the FO-tuple of the adjoint (S*1 ; .... ; S*n-1; P*). [ABSTRACT FROM AUTHOR]