Journal of Operator Theory

Volume 93, Issue 2, Spring 2025 pp. 435-476.

A strictly weakly hypercyclic operator with a hypercyclic subspace

Authors: Kit C. Chan (1), Zeno Madarasz (2)
Author institution: (1) Department of Mathematics and Statistics, Bowling\break Green State University, Bowling Green, OH, 43403, U.S.A.
(2) Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH, 43403, U.S.A.

Summary: An interesting topic of study for a hypercyclic operator $T:X\to X$ on a topological vector space $X$ has been whether $X$ has an infinite\hyp{}dimensional, closed subspace consisting entirely, except for the zero vector, of hypercyclic vectors of $T$. These subspaces are called hypercyclic subspaces. It has been known that there is an operator $T:H\to H$ on a Hilbert space $H$ that is hypercyclic with respect to the weak topology, but fails to be hypercyclic with respect to the norm topology. However, the natural question of whether such an operator $T$ can have a hypercyclic subspace remains unsolved. In the present paper, we answer this question in the positive. Furthermore, we show that a continuous linear transformation $L:B(H)\to B(H)$ on the operator space $B(H)$ can be hypercyclic with respect to the weak operator topology, but fail to be hypercyclic with respect to the strong operator topology.

DOI: http://dx.doi.org/10.7900/jot.2023jun06.2433
Keywords: hypercyclic operator, hypercyclic subspace, weak topology, weak operator topology, strong operator topology

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