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On the supercyclicity and hypercyclicity of the operator algebra. (English) Zbl 1154.47004

Let \(H\) be a separable infinite-dimensional Hilbert space. The \(*\)-strong operator topology on the space \(B(H)\) of all the bounded operators on \(H\) is defined by the family of continuous seminorms \(q_h(T):=\| Th\| +\| T^* h\| \) as \(h\) varies in \(H\). This topology is coarser than the operator norm on \(B(H)\) and finer than the strong operator topology. In this paper, the authors obtain sufficient conditions to ensure that a bounded linear map \(L\) on \(B(H)\) is supercyclic or hypercyclic for the \(*\)-strong operator topology. These results are then applied to left multiplication maps \(L_T(S):=TS\), \(S \in B(H)\). Analogous results for the strong operator topology were discovered by K.C.Chan [J. Oper.Theory 42, No.2, 231–244 (1999; Zbl 0997.47058)], whose ideas are exploited in the paper under review.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47L10 Algebras of operators on Banach spaces and other topological linear spaces

Citations:

Zbl 0997.47058
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References:

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