Frequently hypercyclic subspaces
In [\textit{L. B. González} and \textit{A. Montes Rodríguez}, J. Approx. Theory 82, No. 3, 375--391 (1995; Zbl 0831.30024)], the authors started a new line of investigation by asking if an operator can possess an infinite-dimensional closed subspace all of whose non-zero vectors are hypercyclic, that is, vectors with a dense orbit. In [\textit{A. Montes-Rodríguez}, Mich. Math. J. 43, No. 3, 419--436 (1996; Zbl 0907.47023)], the author obtained a sufficient condition for an operator to have such a subspace, while in the Banach space setting, \textit{M. González} et al. [Proc. Lond. Math. Soc. (3) 81, No. 1, 169--189 (2000; Zbl 1028.47007)] even obtained a full characterization. In the present paper, the authors study the analogous problem for frequently hypercyclic operators, that is, they study the existence of operators with an infinite-dimensional closed subspace all of whose non-zero vectors are frequently hypercyclic. Such a subspace is called a frequently hypercyclic subspace. An operator \(T\) on a separable topological vector space \(X\) is said to be frequently hypercyclic if there is some vector \(x\in X\) such that, for any non-empty open subset \(U\) of \(X,\) the set \(\{n\in \mathbb{N}: \;T^nx\in U\}\) has positive lower density; the lower density of a subset \(A\subseteq \mathbb{N}\) is defined as \(\liminf_{N\rightarrow \infty}\frac{1}{N}\text{card}\{n\leq N: n\in A\}\). The authors give sufficient conditions by attacking the problem with the three methods that have been used for hypercyclicity: a constructive approach, an approach via left-multiplication operators, and an approach via tensor products. More concretely, for \(T\) an operator on a separable \(F\)-space \(X\) with a continuous norm, they add a strong demand on the Frequently Hypercyclicity Criterion (see [\textit{F. Bayart} and \textit{S. Grivaux}, C. R., Math., Acad. Sci. Paris 338, No. 9, 703--708 (2004; Zbl 1059.47006); Trans. Am. Math. Soc. 358, No. 11, 5083--5117 (2006; Zbl 1115.47005)]; \textit{A. Bonilla} and \textit{K. G. Grosse-Erdmann} [Ergodic Theory Dyn. Syst. 27, No. 2, 383--404 (2007; Zbl 1119.47011); erratum ibid. 29, No. 6, 1993--1994 (2009)] in order to find the frequently hypercyclic subspace. This stronger condition is the existence of an infinite-dimensional closed subspace \(M_0\) of \(X\) such that \(T^nx\rightarrow 0\) for all \(x\in M_0\). The authors also give some examples of operators with frequently hypercyclic subspaces and they state some open problems, such as the characterization of these subspaces in the Banach setting, or the existence of frequently hypercyclic operators with a hypercyclic subspace but not a frequently hypercyclic one.
scientific article; zbMATH DE number 6111281
scientific article