Abstract
A bounded linear operator T on a Banach space X is called subspace-hypercyclic if there is a subspace \(M \subsetneq X\) and a vector \(x \in X\) such that \({{\,\mathrm{orb}\,}}{(x,T)} \cap M\) is dense in M. We show that every Banach space supports subspace-hypercyclic operators and, if the space is separable, the operator obtained is also weakly mixing. Additionally, we provide a new criteria for subspace-hypercyclicity, generalizing a previous result from Le.
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Notes
See [5, Corollary 8.3].
In fact, if \(m \in M\) and \(\ker ^*(S) \subseteq M\) is dense in M, then there is \((x_n)_{n \ge 1} \in \ker ^*(S)\) such that \(x_n \rightarrow m\). Since \(S(x_n) \in \ker ^*(S)\) for every \(n \ge 1\) and \(S(x_n) \rightarrow S(m)\) then \(S(m) \in \overline{\ker ^*(S)} = M\).
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Acknowledgements
The first author was supported by CNPq, Grant 142035/2018-1.
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Communicated by Roman Drnovsek.
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Augusto, A., Pellegrini, L. On the existence of subspace-hypercyclic operators and a new criteria for subspace-hypercyclicity. Adv. Oper. Theory 5, 1814–1824 (2020). https://doi.org/10.1007/s43036-020-00095-1
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DOI: https://doi.org/10.1007/s43036-020-00095-1