\(n\)-supercyclic and strongly \(n\)-supercyclic operators in finite dimensions (Q2872217)

A linear and continuous operator \(T\) on a Banach space \(X\) is said to be \textit{supercyclic} if there exists \(x\in X\) such that, if \(E=\operatorname{span} \{x\}\), the orbit of \(E\) under \(T\) is dense in \(X\), i.e., NEWLINE\[NEWLINE\overline{\{T^nx\mid n\in\mathbb{N},\, x\in E\}}^X=X.NEWLINE\]NEWLINE Supercyclicity was introduced by \textit{H. M. Hilden} and \textit{L. J. Wallen} in [Indiana Univ. Math. J. 23, 557--565 (1974; Zbl 0274.47004)]. Several generalizations have been introduced since then, see, for instance, the one by \textit{N. S. Feldman} [Stud. Math. 151, No. 2, 141--159 (2002; Zbl 1006.47008)] who introduced \(n\)-supercyclic operators. \(T\) is said to be \(n\)-supercyclic if there exists an \(n\)-dimensional vector space \(E\) of \(X\) whose orbit under \(T\) is dense in \(X\). \textit{P. S. Bourdon}, \textit{N. S. Feldman} and \textit{J. H. Shapiro} [Stud. Math. 165, No.~2, 135--157 (2004; Zbl 1056.47008)] proved that, in the complex case, this is just an infinite-dimensional phenomenon. \textit{G. Herzog} [Stud. Math. 103, No. 3, 295--298 (1992; Zbl 0811.47018)] stated an analogous result for the real case, proving that there are no \(1\)-supercyclic operators on \(\mathbb{R}^n\) for \(n\geq 3\).NEWLINENEWLINEFrom the author's abstract: ``In this paper, it is proved that, on \(\mathbb{R}^N\), there is no \(n\)-supercyclic operator with \(1\leq n< \lfloor (N+1)/2 \rfloor\), i.e., if \(\mathbb{R}^N\) has an \(n\)-dimensional subspace whose orbit under \(T\in L(\mathbb{R}^N)\) is dense in \(\mathbb{R}^N\), then \(n\) is greater than \(\lfloor (N+1)/2 \rfloor\). Moreover, this value is optimal.''NEWLINENEWLINEIn addition, the author considers also what happens with strongly \(n\)-supercyclic operators on finite-dimensional spaces. This notion of strongly \(n\)-supercyclic was recently introduced by \textit{S. Shkarin} in [J. Math. Anal. Appl. 348, No. 1, 193--210 (2008; Zbl 1148.47009)].NEWLINENEWLINEFrom the author's abstract: ``A linear and continuous operator \(T\in L(\mathbb{R}^N)\) is strongly \(n\)-supercyclic if \(\mathbb{R}^N\) has an \(n\)-dimensional subspace whose orbit under \(T\) is dense in \(\mathbb{P}_n(\mathbb{R}^N)\), the \(n\)-th Grassmannian. It is proved that strong \(n\)-supercyclicity does not occur non-trivially in finite dimensions.''