Automorphisms and the fundamental operators associated with the symmetrized tridisc

TL;DR

This paper derives explicit formulas for automorphisms of the symmetrized polydisc and explores the associated fundamental operator tuples, revealing their structure and conditions for unitary equivalence in the context of operator theory.

Contribution

It provides explicit formulas for automorphisms in their own coordinates and characterizes the fundamental operator tuples for $ ext{symmetrized polydisc}$ automorphisms, including non-commuting cases.

Findings

Explicit automorphism formulas for $ ext{symmetrized polydisc}$

Characterization of fundamental operator tuples for automorphisms

Conditions for unitary equivalence of $ ext{Gamma}_n$-contractions

Abstract

The automorphisms of the symmetrized polydisc $\mathbb G_n$ are well-known and are given in the coordinates of the polydisc in \cite{E:Z}. 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 \[ S_i-S_{n-i}^*P=D_PA_iD_P\,, \quad D_P=(I-P^*P)^{1/2}\,, \] 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 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.