Power regularity of $d$-tuple of operators

A bounded linear operator $S$ on a Hilbert space $\mathcal{H}$ is power regular if $\lim_{n\rightarrow \infty}\Vert S^{n}x\Vert^{1/n}$ exists for every $x\in \mathcal {H}$. In this talk, we define the concept of power regularity for commuting tuples of opertors and prove that the spherical $m$-isometries are power regular. Moreover, we provide conditions on a right invertible spherical isometry making it a spherical unitary.