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 (S 1 , ⋯ , 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 (S 1 , ⋯ , S n - 1 , P) , there exist n - 1 unique operators A 1 , ⋯ , A n - 1 such that S i - S n - i ∗ P = D P A i D P , D P = (I - P ∗ P) 1 / 2 , <graphic href="12044_2020_598_Article_Equ4.gif"></graphic> for i = 1 , ⋯ , n - 1 . This unique (n - 1) -tuple (A 1 , ⋯ , A n - 1) , which is called the fundamental operator tuple or F O -tuple of (S 1 , ⋯ , 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 (S 1 , ⋯ , 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. [ABSTRACT FROM AUTHOR]