On supercyclicity of tuples of operators (Q745939)

This article continues work by \textit{N. S. Feldman} in [J. Math. Anal. Appl. 346, No. 1, 82--98 (2008; Zbl 1148.47008)], and presents several results about supercyclic and hypercyclic \(n\)-tuples \(T=(T_1,\dots,T_n)\) of commuting bounded operators on a separable Banach space \(X\). The following results are shown: {\parindent=6mm \begin{itemize}\item[(1)] There are no supercyclic normal tuples if \(X\) is an infinite dimensional Hilbert space. \item[(2)] If the Harte spectrum of a hypercyclic \(n\)-tuple is not empty, then it intersects the complement of the unit polydisc of \(\mathbb{C}^n\). Moreover, it might happen that the Harte spectrum does not intersect the closure of the unit polydisc. \item[(3)] There are no hypercyclic \(n\)-tuples of diagonal matrices in \(\mathbb{C}^n\), but supercyclicity might occur. \item[(4)] If \(T\) is a supercyclic \(\ell\)-tuple of commuting \(n \times n\) complex matrices, then \(\ell \geq n\). \item[(5)] If \(T=(T_1,\dots,T_n)\) is supercyclic, then the matrices \(T_j\)'s are simultaneously diagonalizable. \end{itemize}}