HYPERCYCLICITY OF THE OPERATOR ALGEBRA FOR A SEPARABLE HILBERT SPACE

Abstract

Abstract. 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 onH to have a closed infinite dimensional subspace of hypercyclic vectors. This condition was first found by Montes-Rodŕıguez for a general Banach space, but the approach that we take is entirely different and simpler.