## On the dynamics of the $$d$$-tuples of $$m$$-isometries.(English)Zbl 07036628

Summary: 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 $$Q_T(A)=\sum_{i=1}^dT_i^\ast AT_i$$ for every bounded linear operator $$A$$ on $$\mathcal{H}$$. Also, $$T$$ is called a toral $$m$$-isometry if $$\sum_{p\in\mathbb{N}^{d},\,0\leq p\leq n}(-1)^{|p|}\binom{n}{p}{T^\ast}^p T^p=0$$ for all $$n\in\mathbb{N}^d$$ with $$|n|=m$$. 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.

### MSC:

 47A13 Several-variable operator theory (spectral, Fredholm, etc.) 47A16 Cyclic vectors, hypercyclic and chaotic operators 47B47 Commutators, derivations, elementary operators, etc.
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