Automorphisms and the fundamental operators associated with the symmetrized tridisc

Abstract

The automorphisms of the symmetrized polydisc $${\mathbb {G}}_n$$ 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 $${\mathbb {G}}_n$$ in its own coordinates. If $$\tau $$ is an automorphism of $${\mathbb {G}}_n$$ , then $$\tau (S_1,\dots ,S_{n-1},P)$$ is a $$\Gamma _n$$ -contraction, where a $$\Gamma _n$$ -contraction is a commuting n-tuple of Hilbert space operators for which the closed symmetrized polydisc $$\Gamma _n$$ is a spectral set. Corresponding to every $$\Gamma _n$$ -contraction $$(S_1,\dots ,S_{n-1},P)$$ , there exist $$n-1$$ unique operators $$A_1,\dots ,A_{n-1}$$ such that $$\begin{aligned} \qquad \qquad \qquad S_i-S_{n-i}^*P=D_PA_iD_P, \quad D_P=(I-P^*P)^{1/2}, \end{aligned}$$ for $$i=1,\dots , n-1$$ . This unique $$(n-1)$$ -tuple $$(A_1,\dots ,A_{n-1})$$ , which is called the fundamental operator tuple or $${\mathcal {F}}_O$$ -tuple of $$(S_1,\dots ,S_{n-1},P)$$ in the literature, plays central role in every section of operator theory on $$\Gamma _n$$ . We find an explicit form of the $${\mathcal {F}}_O$$ -tuple of $$\tau (S_1,\dots ,S_{n-1},P)$$ when $$n=3$$ . We show by an example that a $$\Gamma _n$$ -contraction may not have commuting $${\mathcal {F}}_O$$ -tuple. Also, we obtain a necessary and sufficient condition under which two $$\Gamma _n$$ -contractions are unitarily equivalent.