Automorphisms and the fundamental operators associated with the symmetrized tridisc

The automorphisms of the symmetrized polydisc G_n are well-known and are given in the coordinates of the polydisc in Edigarian and Zwonek (Arch. Math.84 (2005) 364–374). We find an explicit formula for the automorphisms of G_n in its own coordinates. If τ is an automorphism of G_n, then τ(S₁, ⋯ , S_n-1, P) is a Γ _n-contraction, where a Γ _n-contraction is a commuting n-tuple of Hilbert space operators for which the closed symmetrized polydisc Γ _n is a spectral set. Corresponding to every Γ _n-contraction (S₁, ⋯ , S_n-1, P) , there exist n- 1 unique operators A₁, ⋯ , A_n-1 such that Si-Sn-i∗P=DPAiDP,DP=(I-P∗P)1/2,for i= 1 , ⋯ , n- 1. This unique (n- 1) -tuple (A₁, ⋯ , A_n-1) , which is called the fundamental operator tuple or F_O-tuple of (S₁, ⋯ , S_n-1, P) in the literature, plays central role in every section of operator theory on Γ _n. We find an explicit form of the F_O-tuple of τ(S₁, ⋯ , S_n-1, P) when n= 3. We show by an example that a Γ _n-contraction may not have commuting F_O-tuple. Also, we obtain a necessary and sufficient condition under which two Γ _n-contractions are unitarily equivalent.