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Semigroups of chaotic operators. (English) Zbl 1182.47007
Let $X$ be an infinite-dimensional separable Banach space. A linear and continuous operator $T\in L(X)$ is said to be {\it hypercyclic} if there exists some $x\in X$ whose orbit under $T$ is dense. If, in addition, the set of periodic points for $T$ is dense, then $T$ is said to be {\it chaotic}. Analogous definitions can also be stated for a $C_0$-semigroup $\{T_t\}_{t\geq 0}$ of linear and continuous operators in $L(X)$. There are some differences in the treatment of hypercyclicity and chaos: It is known that every infinite-dimensional separable Banach space supports a hypercyclic operator [see {S. I. Ansari}, J. Funct. Anal. 148, No. 2, 384--390 (1997; Zbl 0898.47019) and {\it L. Bernal-González}, Proc. Am. Math. Soc. 127, No. 4, 1003--1010 (1999; Zbl 0911.47020); see also {\it J. Bonet} and {\it A. Peris}, J. Funct. Anal. 159, No. 2, 587--595 (1998; Zbl 0926.47011) for the Fréchet case]. However, the counterpart for chaotic operators is not fulfilled [{\it J. Bonet, F. Martínez-Giménez} and {\it A. Peris}, Bull. Lond. Math. Soc. 33, No. 2, 196--198 (2001; Zbl 1046.47008)]. In both cases, analogous results can also be stated for $C_0$-semigroups [{\it T. Bermúdez, A. Bonilla} and {\it A. Martinón}, Proc. Am. Math. Soc. 131, No. 8, 2435--2441 (2003; Zbl 1044.47006)]. The hypercyclicity is preserved if we restrict ourselves to sub-semigroups: {\it S. I.\thinspace Ansari} proved that if $T$ is hypercyclic, then $T^p$ is hypercyclic for every $p\in\mathbb{N}$ [J. Funct. Anal. 128, No. 2, 374--383 (1995; Zbl 0853.47013)]. On the other hand, {\it F. León-Saavedra} and {\it V. Müller} proved that the operator $T$ is hypercyclic if and only if $\lambda T$ is hypercyclic for every $\lambda\in\mathbb{C}$ with $|\lambda|=1$ [Integral Equations Oper. Theory 50, No. 3, 385--391 (2004; Zbl 1079.47013)]. The ideas in this paper gave the key to prove that all nontrivial operators on a hypercyclic $C_0$-semigroup $\{T_t\}_{t\geq 0}$ are also hypercyclic [{\it J. A.\thinspace Conejero, V. Müller} and {\it A. Peris}, J. Funct. Anal. 244, No. 1, 342--348 (2007; Zbl 1123.47010)]. In the paper under review, the authors give a negative answer to the problem of extending these last two results to the chaotic setting. This gives a further insight of the relations between the dynamical properties of a $C_0$-semigroup and its operators. The authors give an example of a chaotic operator $T$ such that $\lambda T$ is not chaotic for certain $\lambda\in\mathbb{C}$ with $|\lambda|=1$. Besides, they give an example of a chaotic $C_0$-semigroup $\{T_t\}_{t\geq 0}$ such that there exists $t_0,t_1\neq 0$ such that $T_{t_0}$ is chaotic and $T_{t_1}$ is not chaotic. Even more, they construct a chaotic $C_0$-semigroup that does not contain any chaotic operator. The constructions are given from a detailed study of the point spectrum of chaotic operators.

MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators
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