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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 TL(X) is said to be hypercyclic if there exists some xX 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 {T t } t0 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 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 [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 λT is hypercyclic for every λ with |λ|=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 } t0 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 λT is not chaotic for certain λ with |λ|=1. Besides, they give an example of a chaotic C 0 -semigroup {T t } t0 such that there exists t 0 ,t 1 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:
47A16Cyclic vectors, hypercyclic and chaotic operators