Characteristic functions for multicontractions and automorphisms of the unit ball

A \emph{multicontraction} on a Hilbert space $\text{HH}$ is an $n$-tuple of operators $T=(T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_n)$ acting on $\text{HH}$, such that $\sum_{i=1}^n T_i T_i^*\le \1_\HH$. We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball.