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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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