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 \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.

13 Pages

References in corpus (1)

• A Schwarz lemma for the symmetrized tridisc and description of interpolating functions