Abstract
This article introduces and studies various supercyclicity criteria for \(C_0\)-semigroups. It is shown that all the different versions are equivalent and several examples are given.
Similar content being viewed by others
References
Abate, M.: Iteration Theory of Holomorphic Maps on Taut Manifolds. Mediterranean Press, Rende (1989)
Bayart, F., Matheron, E.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)
Berkson, E., Porta, H.: Semigroups of analytic functions and composition operators. Mich. Math. J. 25, 101–115 (1978)
Bermudez, T., Miller, V.G.: On operators T such that f(T) is hypercyclic. Integral Eq. Oper. Theory 37, 332–340 (2000)
Betsakos, D.: Geometric description of the classification of holomorphic semigroups. Proc. Am. Math. Soc. 144, 1595–1604 (2016)
Bourdon, P.S., Shapiro, J.H.: Cyclic composition operators on \(H^2\). Proc. Symp. Pure Math. 51, 43–53 (1990)
Conejero, J.A., Mangino, E.M.: Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators. Mediterr. J. Math. 7, 101–109 (2010)
Cowen, C.C., MacCluer, B.D.: Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)
Desch, W., Schappacher, W., Webb, G.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17, 793–819 (1997)
Desch, W., Schappacher, W.: Hypercyclicity of semigroups is a very unstable property. Math. Model. Nat. Phenom. 3, 148–160 (2008)
Engel, K.J., Nagel, R.: One-parameter semigroups for linear evolution equations. Springer-Verlag, Berlin (2000)
Gallardo-Gutiérrez, E.A., Montes-Rodríguez, A.: The role of the spectrum in the cyclic behavior of composition operators. Mem. Amer. Math. Soc. 167, 791 (2004)
Grosse-Erdmann, K.G., Manguillot, A.P.: Linear Chaos. Universitext. Springer, London (2011)
Herrero, D.A.: Limits of hypercyclic and supercyclic operators. J. Funct. Anal. 99, 179–190 (1991)
Kalmas, T.: Hypercyclic, mixing, and chaotic \(C_0\)-semigroups. Ph.D. Thesis, (2006)
Kumar, A., Srivastava, S.: Supercyclic \(C_0\)-semigroups, stability and somewhere dense orbits. J. Math. Anal. Appl. 476, 539–548 (2019)
Matsui, M., Yamada, M., Takeo, F.: Supercyclic and chaotic translation semigroups. Proc. Am. Math. Soc. 131, 3535–3546 (2003)
Mourchid, S.E.: On a hypercyclicity criterion for strongly continuous semigroups. Discr. Contin. Dyn. Syst. 13, 271–275 (2005)
Rodriguez, A.M., Salas, H.N.: Supercyclic subspaces: spectral theory and weighted shifts. Adv. Math. 163, 74–134 (2001)
De Rosa, M., la Read, C.: A hypercyclic operator whose direct sum \(T \oplus T\) is not hypercyclic. J. Oper. Theor. 61, 369–380 (2009)
Shapiro, J.H.: Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York (1993)
Shkarin, S.: On supercyclicity of operators from a supercyclic semigroup. J. Math. Ann. Appl. 282, 516–522 (2011)
Siskakis, A.G.: Semigroups of composition operators on spaces of analytic functions, a review. Contemp. Math. 213, 229 (1998)
Takeo, F.: Chaos and hypercyclicity for solution semigroups to some partial differential equations. Nonlinear Anal. 63, 1943–1953 (2005)
Teresa, B., Bonilla, A., Peris, A.: On hypercyclicity and supercyclicity criteria. Bull. Austral. Math. Soc. 70, 45–54 (2004)
Acknowledgements
The authors are grateful to the reviewers for their comments and suggestions that helped in greatly improving this paper. The second author was supported by an R&D grant from the University of Delhi, India.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Klaus Thomsen.
Rights and permissions
About this article
Cite this article
Kumar, A., Srivastava, S. Supercyclicity criteria for \(C_0\)-semigroups. Adv. Oper. Theory 5, 1646–1666 (2020). https://doi.org/10.1007/s43036-020-00073-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43036-020-00073-7