International Journal of Nonlinear Analysis and Applications

On ‎$‎J‎$‎-class $C_0$-semigroups of operators

Document Type : Research Paper

Authors

1 Department of Pure Mathematics‎, ‎Ferdowsi University‎ of Mashhad‎, ‎International Campus‎, ‎Mashhad‎, ‎Iran

2 Department of Pure Mathematics‎, ‎Ferdowsi University‎ ‎of Mashhad‎, ‎P‎. ‎O‎. ‎Box 1159‎, ‎Mashhad 91775‎, ‎Iran‎

Abstract

‎In this paper‎, ‎locally topologically transitive (or J-class) $C_0$-semigroups of operators on Banach spaces are studied‎. ‎Some similarity and differences of locally transitivity and hypercyclicity of $C_0$-semigroups are investigated‎. ‎Next the Kato's limit of a sequence of $C_0$-semigroups are considered and their locally transitivity relations are studied‎.

Keywords

[1] T. Bermudez, A. Bonilla and A. Martin´on, On the existence of chaotic and hyper cyclic semigroups on Banach spaces, Proc. Amer. Math. Soc. 131(8) (2003) 2435–2441.
[2] T. Berm´urdez, A. Bonilla, J. A. Conejero and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math. 170(1) (2005) 57–75.
[3] G. Costakis and A. Manoussos, J-class weighted shifts on the space of bounded sequences of complex numbers, Integral Equ. Oper. Theory 62 (2008) 149–158.
[4] G. Costakis and A. Manoussos, J-class operators and hypercyclicity, J. Operator Theory, 67 (2012) 101–119.
[5] R. DeLaubenfels, H. Emamirad and V. Protopopescu, Linear chaos and approximation, J. Approx. Theory 105(1) (2000) 176–187.
[6] W. Desch, W. Schappacher and G.F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Syst. 17 (1997) 793–819.
[7] H. Emamirad, Hypercyclicity in the scattering theory for linear transport equation, Trans. Amer. Math. Soc. 350 (1998) 3707–3716.
[8] K.J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
[9] R. Gethner and J.H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100(2) (1987) 281–288.
[10] T. Kalmes, On chaotic C0-semigroups and infinitely regular hyper cyclic vectors, Proc. Amer. Math. Soc. 134 (2006) 2997–3002.
[11] T. Kalmes, Hypercyclic, Mixing, and Chaotic C0-Semigroups, Ph. D. Thesis, Trier University, 2006.
[12] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966.
[13] C. Kitai, Invariant closed sets for linear operators, Ph. D. Thesis, University of Toronto, 1982.
[14] A.B. Nasseri, J-class Operators on Certain Banach Spaces, Ph. D. Thesis, Dortmund University, 2013.
[15] A.B. Nasseri, On the existence of J-class operators on Banach spaces, Proc. Amer. Math. Soc. 140 (2012) 3549–3555.
[16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1992.
[17] V. Protopopescu and Y. Azmy, Topological chaos for a class of linear models, Math. Models Methods Appl. Sci. 2 (1992) 79–90.
[18] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969) 17–22.
[19] G. Tian and B. Hou, Limits of J-class operators, Proc. Amer. Math. Soc. 142(5) (2014) 1663–1667.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 397-403
  • Receive Date: 27 August 2017
  • Revise Date: 15 October 2017
  • Accept Date: 29 October 2017