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

Supercyclic translation $C_0$-semigroup on complex sectors

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 $\mathbb{C}$.
Citation: Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 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.

[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. doi: 10.1016/j.jfa.2006.12.008.

[3]

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

[4]

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

[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. doi: 10.1017/S0143385797084976.

[6]

K. G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos,, Springer, (2011). doi: 10.1007/978-1-4471-2170-1.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

S. Rolewicz, On orbits of elements,, Studia Math., 32 (1969), 17.

[16]

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

[17]

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

show all references

References:
[1]

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

[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. doi: 10.1016/j.jfa.2006.12.008.

[3]

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

[4]

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

[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. doi: 10.1017/S0143385797084976.

[6]

K. G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos,, Springer, (2011). doi: 10.1007/978-1-4471-2170-1.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

S. Rolewicz, On orbits of elements,, Studia Math., 32 (1969), 17.

[16]

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

[17]

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

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