Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

Xinxing Wu

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 1, page 105-114
  • ISSN: 0011-4642

Abstract

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During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator B w n : λ p ( A ) → λ p ( A ) defined on the Köthe sequence space λ p ( A ) exhibits distributional ϵ -chaos for any 0 < ϵ < diam λ p ( A ) and any n ∈ ℕ is obtained. Under this assumption, the principal measure of B w n is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional ϵ -chaos for any 0 < ϵ < diam λ p ( A ) .

How to cite

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Wu, Xinxing. "Maximal distributional chaos of weighted shift operators on Köthe sequence spaces." Czechoslovak Mathematical Journal 64.1 (2014): 105-114. <http://eudml.org/doc/262001>.

@article{Wu2014,
abstract = {During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_\{w\}^\{n\}\colon \lambda _\{p\}(A)\rightarrow \lambda _\{p\}(A)$ defined on the Köthe sequence space $\lambda _\{p\}(A)$ exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop \{\rm diam\} \lambda _\{p\}(A)$ and any $n\in \mathbb \{N\}$ is obtained. Under this assumption, the principal measure of $B_\{w\}^\{n\}$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop \{\rm diam\} \lambda _\{p\}(A)$.},
author = {Wu, Xinxing},
journal = {Czechoslovak Mathematical Journal},
keywords = {weighted shift operator; principal measure; distributional chaos; weighted shift operator; principal measure; distributional chaos},
language = {eng},
number = {1},
pages = {105-114},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Maximal distributional chaos of weighted shift operators on Köthe sequence spaces},
url = {http://eudml.org/doc/262001},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Wu, Xinxing
TI - Maximal distributional chaos of weighted shift operators on Köthe sequence spaces
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 105
EP - 114
AB - During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_{w}^{n}\colon \lambda _{p}(A)\rightarrow \lambda _{p}(A)$ defined on the Köthe sequence space $\lambda _{p}(A)$ exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop {\rm diam} \lambda _{p}(A)$ and any $n\in \mathbb {N}$ is obtained. Under this assumption, the principal measure of $B_{w}^{n}$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop {\rm diam} \lambda _{p}(A)$.
LA - eng
KW - weighted shift operator; principal measure; distributional chaos; weighted shift operator; principal measure; distributional chaos
UR - http://eudml.org/doc/262001
ER -

References

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  1. Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A., 10.1016/j.jmaa.2010.06.011, J. Math. Anal. Appl. 373 (2011), 83-93. (2011) Zbl1214.47012MR2684459DOI10.1016/j.jmaa.2010.06.011
  2. Duan, J., Fu, X.-C., Liu, P.-D., Manning, A., 10.1016/S0893-9659(98)00119-0, Appl. Math. Lett. 12 (1999), 15-19. (1999) Zbl0956.37057MR1663405DOI10.1016/S0893-9659(98)00119-0
  3. Köthe, G., Topological Vector Spaces I, Translated from German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159 Springer, New York (1969). (1969) MR0248498
  4. Li, T. Y., Yorke, J. A., 10.2307/2318254, Am. Math. Mon. 82 (1975), 985-992. (1975) Zbl0351.92021MR0385028DOI10.2307/2318254
  5. Martínez-Giménez, F., Oprocha, P., Peris, A., 10.1016/j.jmaa.2008.10.049, J. Math. Anal. Appl. 351 (2009), 607-615. (2009) Zbl1157.47008MR2473967DOI10.1016/j.jmaa.2008.10.049
  6. Martínez-Giménez, F., 10.1090/S0002-9939-07-08658-3, Proc. Am. Math. Soc. 135 (2007), 1741-1752. (2007) Zbl1115.47008MR2286084DOI10.1090/S0002-9939-07-08658-3
  7. Meise, R., Vogt, D., Introduction to Functional Analysis, Translated from the German by M. S. Ramanujan and revised by the authors. Oxford Graduate Texts Mathematics 2 The Clarendon Press, Oxford University Press, New York (1997). (1997) Zbl0924.46002MR1483073
  8. Oprocha, P., Wilczyński, P., 10.1016/j.chaos.2005.09.069, Chaos Solitons Fractals 31 (2007), 347-355. (2007) Zbl1140.37303MR2259760DOI10.1016/j.chaos.2005.09.069
  9. Pikula, R., 10.4064/cm107-2-1, Colloq. Math. 107 (2007), 167-177. (2007) Zbl1130.37327MR2284159DOI10.4064/cm107-2-1
  10. Schweizer, B., Smítal, J., 10.1090/S0002-9947-1994-1227094-X, Trans. Am. Math. Soc. 344 (1994), 737-754. (1994) Zbl0812.58062MR1227094DOI10.1090/S0002-9947-1994-1227094-X
  11. Schweizer, B., Sklar, A., Smítal, J., 10.2307/44154056, Real Anal. Exch. 26 (2000/01), 495-524. (2000) MR1844132DOI10.2307/44154056
  12. Smítal, J., Štefánková, M., 10.1016/j.chaos.2003.12.105, Chaos Solitons Fractals 21 (2004), 1125-1128. (2004) Zbl1060.37037MR2047330DOI10.1016/j.chaos.2003.12.105
  13. Wang, L., Huan, S., Huang, G., 10.1016/j.na.2006.12.048, Nonlinear Anal., Theory Methods Appl. 68 (2008), 1682-1686. (2008) Zbl1142.37309MR2388841DOI10.1016/j.na.2006.12.048
  14. Wu, X., Zhu, P., 10.1016/j.aml.2011.09.055, Appl. Math. Lett. 25 (2012), 545-549. (2012) Zbl1242.37009MR2856030DOI10.1016/j.aml.2011.09.055
  15. Wu, X., Zhu, P., The principal measure of a quantum harmonic oscillator, J. Phys. A, Math. Theor. 44 (2011), ID 505101, 6 pages. (2011) Zbl1238.81129MR2869648
  16. Wu, X., Zhu, P., Chaos in a class of nonconstant weighted shift operators, Int. J. Bifurcation Chaos Appl. Sci. Eng. 23 (2013), ID 1350010, 9 pages. (2013) Zbl1270.37016MR3038635

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