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January  2016, 36(1): 361-370. doi: 10.3934/dcds.2016.36.361

Supercyclic translation $C_0$-semigroup on complex sectors

Yu-Xia Liang 1, and  Ze-Hua Zhou 2,

1. 

School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
2. 

Department of Mathematics, Tianjin University, Tianjin 300072, China

Received  May 2013 Revised  April 2015 Published  June 2015

We characterize the supercyclic behavior of sequences of operators in a $C_0$-semigroup whose index set is a sector $\Delta$ in the complex plane $\mathbb{C}$.
Keywords: translation semigroups., Supercyclic.
Mathematics Subject Classification: Primary: 37D45; Secondary: 47A16, 47D6.
Citation: Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361
References:
[1]

F. Bayart and E. Matheron, Dynamics of Linear Operators, Camberidge University Press, 2009. doi: 10.1017/CBO9780511581113.  Google Scholar

[2]

J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal., 244 (2007), 342-348. doi: 10.1016/j.jfa.2006.12.008.  Google Scholar

[3]

J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208. doi: 10.3934/dcds.2009.25.1195.  Google Scholar

[4]

J. A. Conejero and A. Peris, Chaotic translation semigroups, Discrete Contin. Dyn. Syst. Supplement, (2007), 269-276.  Google Scholar

[5]

W. Desch, W. Schappacher and G. Webb., Hypercyclic and chaotic semigroup and chaotic semigroup of linear operators, Ergod. Th. Dynam. Sys., 17 (1997), 793-819. doi: 10.1017/S0143385797084976.  Google Scholar

[6]

K. G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos, Springer, New York, 2011. doi: 10.1007/978-1-4471-2170-1.  Google Scholar

[7]

K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math., 139 (2000), 47-68.  Google Scholar

[8]

G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.  Google Scholar

[9]

D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99 (1991), 179-190. doi: 10.1016/0022-1236(91)90058-D.  Google Scholar

[10]

Y. X. Liang and Z. H. Zhou, Hereditarily hypercyclicity and supercyclicity of different weighted shifts, J. Korean Math. Soc., 51 (2014), 363-382. doi: 10.4134/JKMS.2014.51.2.363.  Google Scholar

[11]

Y. X. Liang and Z. H. Zhou, Hypercyclic behaviour of multiples of composition operators on the weighted Banach space of holomorphic functions, Bull. Belg. Math. Soc. Simon Stevin, 21 (2014), 385-401.  Google Scholar

[12]

Y. X. Liang and Z. H. Zhou, Supercyclic tuples of the adjoint weighted composition operators on Hilbert spaces, Bull. Iranian Math. Soc., 41 (2015), 121-139.  Google Scholar

[13]

M. Matsui, M. Yamada and F. Takeo, Supercyclic and chaotic translation semigroups, Proc. Amer. Math. Soc., 131 (2003), 3535-3546. doi: 10.1090/S0002-9939-03-06960-0.  Google Scholar

[14]

M. Matsui, M. Yamada and F. Takeo, Erratum to"supercyclic and chaotic translation semigroups", Proc. Amer. Math. Soc., 132 (2004), 3751-3752. doi: 10.1090/S0002-9939-04-07608-7.  Google Scholar

[15]

S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.  Google Scholar

[16]

M. Yamada and F. Takeo, Chaotic tranlation semigroups of liear operators, RIMS Koukyroku, 1100 (1999), 8-18.  Google Scholar

[17]

M. Yamada and F. Takeo, Supercyclic tranlation semigroups of liear operators, RIMS Koukyroku, 1186 (2001), 49-56.  Google Scholar

show all references

References:
[1]

F. Bayart and E. Matheron, Dynamics of Linear Operators, Camberidge University Press, 2009. doi: 10.1017/CBO9780511581113.  Google Scholar

[2]

J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal., 244 (2007), 342-348. doi: 10.1016/j.jfa.2006.12.008.  Google Scholar

[3]

J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208. doi: 10.3934/dcds.2009.25.1195.  Google Scholar

[4]

J. A. Conejero and A. Peris, Chaotic translation semigroups, Discrete Contin. Dyn. Syst. Supplement, (2007), 269-276.  Google Scholar

[5]

W. Desch, W. Schappacher and G. Webb., Hypercyclic and chaotic semigroup and chaotic semigroup of linear operators, Ergod. Th. Dynam. Sys., 17 (1997), 793-819. doi: 10.1017/S0143385797084976.  Google Scholar

[6]

K. G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos, Springer, New York, 2011. doi: 10.1007/978-1-4471-2170-1.  Google Scholar

[7]

K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math., 139 (2000), 47-68.  Google Scholar

[8]

G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.  Google Scholar

[9]

D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99 (1991), 179-190. doi: 10.1016/0022-1236(91)90058-D.  Google Scholar

[10]

Y. X. Liang and Z. H. Zhou, Hereditarily hypercyclicity and supercyclicity of different weighted shifts, J. Korean Math. Soc., 51 (2014), 363-382. doi: 10.4134/JKMS.2014.51.2.363.  Google Scholar

[11]

Y. X. Liang and Z. H. Zhou, Hypercyclic behaviour of multiples of composition operators on the weighted Banach space of holomorphic functions, Bull. Belg. Math. Soc. Simon Stevin, 21 (2014), 385-401.  Google Scholar

[12]

Y. X. Liang and Z. H. Zhou, Supercyclic tuples of the adjoint weighted composition operators on Hilbert spaces, Bull. Iranian Math. Soc., 41 (2015), 121-139.  Google Scholar

[13]

M. Matsui, M. Yamada and F. Takeo, Supercyclic and chaotic translation semigroups, Proc. Amer. Math. Soc., 131 (2003), 3535-3546. doi: 10.1090/S0002-9939-03-06960-0.  Google Scholar

[14]

M. Matsui, M. Yamada and F. Takeo, Erratum to"supercyclic and chaotic translation semigroups", Proc. Amer. Math. Soc., 132 (2004), 3751-3752. doi: 10.1090/S0002-9939-04-07608-7.  Google Scholar

[15]

S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.  Google Scholar

[16]

M. Yamada and F. Takeo, Chaotic tranlation semigroups of liear operators, RIMS Koukyroku, 1100 (1999), 8-18.  Google Scholar

[17]

M. Yamada and F. Takeo, Supercyclic tranlation semigroups of liear operators, RIMS Koukyroku, 1186 (2001), 49-56.  Google Scholar

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