Universality and chaos for tensor products of operators.
A bounded operator on a separable Banach space \(\mathcal B\) is said to be hypercyclic if there is a vector \(x \in \mathcal B\) such that the orbit \(\{T^nx\}_{n\geq 0}\) is dense in \(\mathcal B\). In particular, hypercyclic operators are cyclic and their study is especially interesting in connection with invariant subspaces in the Hilbert space setting. \textit{K. C. Chan} [J. Oper. Theory 42, 231--244 (1999; Zbl 0997.47058)] considered hypercyclicity for continuous linear maps defined on the space \(\mathcal L (\mathcal H)\) (where \(\mathcal H\) is a separable Hilbert space) endowed with the strong operator topology. He proved that if a continuous linear map \(\Lambda: \mathcal L(\mathcal H) \to \mathcal L(\mathcal H)\) is under some hypotheses similar to the well-known hypercyclicity criterion, then \(\Lambda\) is hypercyclic. Chan's result is not a mere generalization. Indeed, Chan found some interesting applications to the usual concept of hypercyclicity on Hilbert spaces. For instance, he gave a simpler proof of a result by the reviewer related to hypercyclic subspaces [Mich. Math. J. 43, 419--436 (1996; Zbl 0907.47023)]. Independently (as it can be checked from the submission dates), \textit{K. C. Chan} and \textit{R. D. Taylor} [Integral Equations Oper. Theory 41, 381--388 (2001; Zbl 0995.46014)] and \textit{M. C. Romero-Moreno} and the reviewer [Stud. Math. 150, 201--213 (2002; Zbl 1006.47009)] extended Chan's methods to the operator algebra of a Banach space. The authors of the paper under review generalize Chan's methods from Banach spaces to tensor products on locally convex spaces. They do this for universal sequences, that is, the sequence of powers of an operator is replaced by any sequence of bounded operators. The concept of universality is rather general and one cannot expect much structure in the theory [\textit{K.--G. Grosse-Erdmann}, Bull. Am. Math. Soc., New Ser. 36, 345--381 (1999; Zbl 0933.47003)]. In Section 1 of the paper under review, the authors give a proof of a result, reminiscent of the Universality Criterion, that provides sufficient conditions on a sequence of tensor products of two sequences of operators acting on locally convex spaces to be universal. In Section 2, they turn their attention to apply their theorem to generalize some known results. For instance, Theorem 2.3 in the paper under review is a generalization to locally convex spaces of the corresponding theorem for hypercyclicity of left multiplication operators in the operator algebra of a Banach space in Chan and Taylor's article (Theorem 2.3) or in Moreno--Romero and the reviewer's article (Theorem 3.2). The paper contains other rather general results, in which the authors use the techniques developed by previous authors to get results on the hypercyclicity of tensor products of operators.
scientific article; zbMATH DE number 2005935
scientific article