zbMATH — the first resource for mathematics

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.

47L30 Abstract operator algebras on Hilbert spaces
47B48 Linear operators on Banach algebras
47A16 Cyclic vectors, hypercyclic and chaotic operators