Journal of Operator Theory
Volume 42, Issue 2, Fall 1999 pp. 231-244.
Hypercyclicity of the operator algebra for a separable Hilbert spaceAuthors: 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→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 {Tnf:n≥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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