Journal of Operator Theory

Volume 42, Issue 2, Fall 1999 pp. 231-244.

Hypercyclicity of the operator algebra for a separable Hilbert space

Authors: Kit C. Chan
Author institution: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403, USA

Summary: If $X$ is a topological vector space and $T: X \rightarrow X$ is a continuous linear mapping, then $T$ is said to be hypercyclic when there is a vector $f$ in $X$ such that the set $\{ T^nf: n \geq 0 \}$ is dense in $X$. When $X$ is a separable Fréchet space, Gethner and Shapiro obtained a sufficient condition for the mapping $T$ to be hypercyclic. In the present paper, we obtain an analogous sufficient condition when $X$ is a particular nonmetrizable space, namely the operator algebra for a separable infinite dimensional Hilbert space $H$, endowed with the strong operator topology. Using our result, we further provide a sufficient condition for a mapping $T$ on $H$ to have a closed infinite dimensional subspace of hypercyclic vectors. This condition was first found by Montes-Rodríguez for a general Banach space, but the approach that we take is entirely different and simpler.

Keywords: Operator algebras, separable Hilbert spaces, strong operator topology, hypercyclic vectors

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