On the dynamics of the $d$-tuples of $m$-isometries

Abstract

A commuting $d$-tuple $T=(T_{1}, \ldots , T_{d})$ of bounded linear operators on a Hilbert space $\mathcal {H}$ is called a spherical $m$-isometry if $\sum _{j=0}^{m}(-1)^{j}\binom {m}{j}Q_{T}^{j}(I)=0$, where $I$ denotes the identity operator and $Q_{T}(A)=\sum _{i=1}^{d}T_{i}^{*}AT_{i}$ for every bounded linear operator $A$ on $\mathcal {H}$. Also, $T$ is called a toral $m$-isometry if $\sum _{p\in \mathbb {N}^{d},\, 0\leq p\leq n}(-1)^{\vert p\vert }\binom {n}{p}{T^{\ast }}^{p}T^{p}=0 $ for all $n\in \mathbb {N}^{d}$ with $\vert n\vert = m$. The present paper mainly focuses on the convex-cyclicity of the $d$-tuples of operators on a separable infinite-dimensional Hilbert space $\mathcal {H}$. In particular, we prove that spherical $m$-isometries are not convex-cyclic. Also, we show that toral and spherical $m$-isometric operators are never supercyclic.