The fundamental operator tuples associated with the symmetrized polydisc

摘要

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 the closure of Ran(I-P*P), such that Si - Sn-i*P=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 determines 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 find some intrinsic properties of them. Given a Γn-contraction (S1,...,Sn-1,P) and n-1 operators A1,..., An-1 defined on the closure of Ran(DP), we provide a necessary and sufficient 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 (S1*,..., Sn-1*,P*).