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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 $T\in L\left(X\right)$ is said to be 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 chaotic. Analogous definitions can also be stated for a ${C}_{0}$-semigroup ${\left\{{T}_{t}\right\}}_{t\ge 0}$ of linear and continuous operators in $L\left(X\right)$.

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 L. Bernal-González, Proc. Am. Math. Soc. 127, No. 4, 1003–1010 (1999; Zbl 0911.47020); see also J. Bonet and 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 [J. Bonet, F. Martínez-Giménez and 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 [T. Bermúdez, A. Bonilla and 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: S. I. Ansari proved that if $T$ is hypercyclic, then ${T}^{p}$ is hypercyclic for every $p\in ℕ$ [J. Funct. Anal. 128, No. 2, 374–383 (1995; Zbl 0853.47013)]. On the other hand, F. León-Saavedra and V. Müller proved that the operator $T$ is hypercyclic if and only if $\lambda T$ is hypercyclic for every $\lambda \in ℂ$ 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 ${\left\{{T}_{t}\right\}}_{t\ge 0}$ are also hypercyclic [J. A. Conejero, V. Müller and 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 $\lambda \in ℂ$ with $|\lambda |=1$. Besides, they give an example of a chaotic ${C}_{0}$-semigroup ${\left\{{T}_{t}\right\}}_{t\ge 0}$ such that there exists ${t}_{0},{t}_{1}\ne 0$ 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