Hypercyclicity of the operator algebra for a separable Hilbert space. (English) Zbl 0997.47058

Chan, Kit C.

J. Oper. Theory 42, No. 2, 231-244 (1999).

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