Universality and chaos for tensor products of operators.
type: Item
instance of: scholarly article
title: Universality and chaos for tensor products of operators.
zbMATH Open document ID: 1062.47014
DOI: 10.1016/S0021-9045(03)00118-7
author: Félix Martínez-Giménez
author: A. Peris
published in: Journal of Approximation Theory
publication date: 2003-11-19T00:00:00Z
review text: 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, …
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46F10
zbMATH DE Number: 2005935
zbMATH Keywords: Fréchet spaces
zbMATH Keywords: universality
zbMATH Keywords: hypercyclic vectors
zbMATH Keywords: tensor products of operators
zbMATH Keywords: chaotic dynamics
reviewed by: Alfonso Montes-Rodriguez
MaRDI profile type: Publication
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full work available at URL: s0021-9045(03)00118-7
OpenAlex ID: W1976143538
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description: scientific article; zbMATH DE number 2005935