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Hypercyclicity of the operator algebra for a separable Hilbert space. (English) Zbl 0997.47058
Summary: If $$X$$ is a topological vector space and $$T\colon X\to X$$ is a continuous linear mapping, then $$T$$ is said to be hypercyclic when there is a vector $$f\in X$$ such that the set $$\{T^n f: n\geq 0\}$$ 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 found by Montes-Rodríguez for a general Banach space, but the approach that we take is entirely different and simpler.

##### MSC:
 47L30 Abstract operator algebras on Hilbert spaces 47B48 Linear operators on Banach algebras 47A16 Cyclic vectors, hypercyclic and chaotic operators
##### Keywords:
operator algebras; hypercyclic vectors