Hypercyclic 𝐶₀-semigroups and evolution families generated by first order differential operators

Kalmes, T.

doi:10.1090/S0002-9939-09-09955-9

Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators

Author: T. Kalmes
Journal: Proc. Amer. Math. Soc. 137 (2009), 3833-3848
MSC (2000): Primary 47A16, 47D06
DOI: https://doi.org/10.1090/S0002-9939-09-09955-9
Published electronically: June 18, 2009
MathSciNet review: 2529893
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that $C_0$-semigroups generated by first order partial differential operators on $L^p(\Omega ,\mu )$ and $C_{0,\rho }(\Omega )$, respectively, are hypercyclic if and only if they are weakly mixing, where $\Omega \subset \mathbb {R}^d$ is open. In the case of $d=1$ we give an easy to check characterization of when this happens. Furthermore, we give an example of a hypercyclic evolution family such that not all of the operators of the family are hypercyclic themselves. This stands in complete contrast to hypercyclic $C_0$-semigroups.

Teresa Bermúdez, Antonio Bonilla, and Antonio Martinón, On the existence of chaotic and hypercyclic semigroups on Banach spaces, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2435–2441. MR 1974641, DOI https://doi.org/10.1090/S0002-9939-02-06762-X
Teresa Bermúdez, Antonio Bonilla, José A. Conejero, and Alfredo Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math. 170 (2005), no. 1, 57–75. MR 2142183, DOI https://doi.org/10.4064/sm170-1-3
José A. Conejero and Alfredo Peris, Linear transitivity criteria, Topology Appl. 153 (2005), no. 5-6, 767–773. MR 2203889, DOI https://doi.org/10.1016/j.topol.2005.01.009
Jose A. Conejero, V. Müller, and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal. 244 (2007), no. 1, 342–348. MR 2294487, DOI https://doi.org/10.1016/j.jfa.2006.12.008

J. A. Conejero, E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators, to appear in Mediterr. J. Math.

George Costakis and Alfredo Peris, Hypercyclic semigroups and somewhere dense orbits, C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 895–898 (English, with English and French summaries). MR 1952545, DOI https://doi.org/10.1016/S1631-073X%2802%2902572-4

Wolfgang Desch, Wilhelm Schappacher, and Glenn F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems 17 (1997), no. 4, 793–819. MR 1468101, DOI https://doi.org/10.1017/S0143385797084976

Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989

Karl-Goswin Große-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), iv+84 (German). Dissertation, University of Trier, Trier, 1987. MR 877464

Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI https://doi.org/10.1090/S0273-0979-99-00788-0

K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 2, 273–286 (English, with English and Spanish summaries). MR 2068180

T. Kalmes, Hypercyclic, mixing, and chaotic $C_0$-semigroups induced by semiflows, Ergodic Theory Dynam. Systems 27 (2007), no. 5, 1599–1631. MR 2358980, DOI https://doi.org/10.1017/S0143385707000144

S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22. MR 241956, DOI https://doi.org/10.4064/sm-32-1-17-22

Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157