Supercyclic translation C_0-semigroup on complex sectors

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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

Abstract
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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 and 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.
[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.
[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.
[4]	J. A. Conejero and A. Peris, Chaotic translation semigroups, Discrete Contin. Dyn. Syst. Supplement, (2007), 269-276.
[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.
[6]	K. G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos, Springer, New York, 2011. doi: 10.1007/978-1-4471-2170-1.
[7]	K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math., 139 (2000), 47-68.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.
[16]	M. Yamada and F. Takeo, Chaotic tranlation semigroups of liear operators, RIMS Koukyroku, 1100 (1999), 8-18.
[17]	M. Yamada and F. Takeo, Supercyclic tranlation semigroups of liear operators, RIMS Koukyroku, 1186 (2001), 49-56.

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), 342-348. 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), 1195-1208. doi: 10.3934/dcds.2009.25.1195.
[4]	J. A. Conejero and A. Peris, Chaotic translation semigroups, Discrete Contin. Dyn. Syst. Supplement, (2007), 269-276.
[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.
[6]	K. G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos, Springer, New York, 2011. doi: 10.1007/978-1-4471-2170-1.
[7]	K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math., 139 (2000), 47-68.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.
[16]	M. Yamada and F. Takeo, Chaotic tranlation semigroups of liear operators, RIMS Koukyroku, 1100 (1999), 8-18.
[17]	M. Yamada and F. Takeo, Supercyclic tranlation semigroups of liear operators, RIMS Koukyroku, 1186 (2001), 49-56.

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