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 {\it 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 Fr\'echet 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-Rodr\'\i guez for a general Banach space, but the approach that we take is entirely different and simpler.

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