Research Article
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Disjoint and simultaneous hypercyclic Rolewicz-type operators

Year 2021, Volume 50, Issue 6, 1609 - 1619, 14.12.2021
https://doi.org/10.15672/hujms.791344

Abstract

We characterize disjoint hypercyclic and supercyclic tuples of unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$, which are a generalization of the unilateral backward shift operator. We show that disjoint hypercyclicity and disjoint supercyclicity are equivalent among a subfamily of these operators and disjoint hypercyclic unilateral Rolewicz-type operators always satisfy the Disjoint Hypercyclicity Criterion. We also characterize simultaneous hypercyclic unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$.

References

  • [1] F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge Tracts in Math- ematics 179. Cambridge University Press, Cambridge, 2009.
  • [2] L. Bernal-González, Disjoint hypercyclic operators, Stud. Math. 182 (2), 113–130, 2007.
  • [3] L. Bernal-González and A. Jung, Simultaneous universality, J. Approx. Theory, 237, 43–65, 2018.
  • [4] J. Bès, Ö. Martin, and R. Sanders, Weighted shifts and disjoint hypercyclicity, J. Operator Theory, 72 (1), 15–40, 2014.
  • [5] J. Bès and A. Peris, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336, 297–315, 2007.
  • [6] D. Bongiorno, U.B. Darji and L. Di Piazza, Rolewicz-type chaotic operators, J. Math Anal. Appl. 431 (1), 518–528, 2015.
  • [7] K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (1), 47–68, 2000.
  • [8] K.-G. Grosse-Erdmann and A. Peris, Linear chaos, Universitext: Tracts in mathe- matics. Springer, New York, 2011.
  • [9] Ö. Martin, Disjoint hypercyclic and supercyclic composition operators, PhD thesis, Bowling Green State University, 2010.
  • [10] Ö. Martin and R. Sanders, Disjoint supercyclic weighted shifts, Integr. Equ. Oper. Theory, 85, 191–220, 2016.
  • [11] S. Rolewicz, On orbits of elements, Studia Math. 32, 17–22, 1969.
  • [12] H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (3), 993–1004, 1995.
  • [13] R. Sanders and S. Shkarin, Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion, J. Math. Anal. Appl. 417, 834-855, 2014.
  • [14] Y.Wang, C. Chen, and Z-H. Zhou, Disjoint hypercyclic weighted pseudoshift operators generated by different shifts, Banach J. Math. Anal. 13 (4), 815–836, 2019.
  • [15] Y. Wang and Y-X Liang, Disjoint supercyclic weighted pseudo-shifts on Banach se- quence spaces, Acta Math. Sci. 39B (4), 1089–1102, 2019.
  • [16] Y. Wang and Z-H. Zhou, Disjoint hypercyclic weighted pseudo-shifts on Banach se- quence spaces, Collect. Math. 69, 437–449, 2018.

Year 2021, Volume 50, Issue 6, 1609 - 1619, 14.12.2021
https://doi.org/10.15672/hujms.791344

Abstract

References

  • [1] F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge Tracts in Math- ematics 179. Cambridge University Press, Cambridge, 2009.
  • [2] L. Bernal-González, Disjoint hypercyclic operators, Stud. Math. 182 (2), 113–130, 2007.
  • [3] L. Bernal-González and A. Jung, Simultaneous universality, J. Approx. Theory, 237, 43–65, 2018.
  • [4] J. Bès, Ö. Martin, and R. Sanders, Weighted shifts and disjoint hypercyclicity, J. Operator Theory, 72 (1), 15–40, 2014.
  • [5] J. Bès and A. Peris, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336, 297–315, 2007.
  • [6] D. Bongiorno, U.B. Darji and L. Di Piazza, Rolewicz-type chaotic operators, J. Math Anal. Appl. 431 (1), 518–528, 2015.
  • [7] K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (1), 47–68, 2000.
  • [8] K.-G. Grosse-Erdmann and A. Peris, Linear chaos, Universitext: Tracts in mathe- matics. Springer, New York, 2011.
  • [9] Ö. Martin, Disjoint hypercyclic and supercyclic composition operators, PhD thesis, Bowling Green State University, 2010.
  • [10] Ö. Martin and R. Sanders, Disjoint supercyclic weighted shifts, Integr. Equ. Oper. Theory, 85, 191–220, 2016.
  • [11] S. Rolewicz, On orbits of elements, Studia Math. 32, 17–22, 1969.
  • [12] H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (3), 993–1004, 1995.
  • [13] R. Sanders and S. Shkarin, Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion, J. Math. Anal. Appl. 417, 834-855, 2014.
  • [14] Y.Wang, C. Chen, and Z-H. Zhou, Disjoint hypercyclic weighted pseudoshift operators generated by different shifts, Banach J. Math. Anal. 13 (4), 815–836, 2019.
  • [15] Y. Wang and Y-X Liang, Disjoint supercyclic weighted pseudo-shifts on Banach se- quence spaces, Acta Math. Sci. 39B (4), 1089–1102, 2019.
  • [16] Y. Wang and Z-H. Zhou, Disjoint hypercyclic weighted pseudo-shifts on Banach se- quence spaces, Collect. Math. 69, 437–449, 2018.

Details

Primary Language English
Subjects Mathematics
Journal Section Mathematics
Authors

Nurhan ÇOLAKOĞLU
Istanbul Technical University
0000-0003-2216-6952
Türkiye


Özgür MARTİN (Primary Author)
MIMAR SINAN FINE ARTS UNIVERSITY
0000-0003-1605-1593
Türkiye

Supporting Institution Mimar Sinan Fine Arts University Scientific Research Project
Project Number 2016-18
Thanks The first author was partially supported by Istanbul Technical University Scientific Research Project [grant no. TAB-2017-40552]. The second author was partially supported by Mimar Sinan Fine Arts University Scientific Research Project [grant no. 2016-18].
Publication Date December 14, 2021
Published in Issue Year 2021, Volume 50, Issue 6

Cite

Bibtex @research article { hujms791344, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2021}, volume = {50}, pages = {1609 - 1619}, doi = {10.15672/hujms.791344}, title = {Disjoint and simultaneous hypercyclic Rolewicz-type operators}, key = {cite}, author = {Çolakoğlu, Nurhan and Martin, Özgür} }
APA Çolakoğlu, N. & Martin, Ö. (2021). Disjoint and simultaneous hypercyclic Rolewicz-type operators . Hacettepe Journal of Mathematics and Statistics , 50 (6) , 1609-1619 . DOI: 10.15672/hujms.791344
MLA Çolakoğlu, N. , Martin, Ö. "Disjoint and simultaneous hypercyclic Rolewicz-type operators" . Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 1609-1619 <https://dergipark.org.tr/en/pub/hujms/issue/66204/791344>
Chicago Çolakoğlu, N. , Martin, Ö. "Disjoint and simultaneous hypercyclic Rolewicz-type operators". Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 1609-1619
RIS TY - JOUR T1 - Disjoint and simultaneous hypercyclic Rolewicz-type operators AU - Nurhan Çolakoğlu , Özgür Martin Y1 - 2021 PY - 2021 N1 - doi: 10.15672/hujms.791344 DO - 10.15672/hujms.791344 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1609 EP - 1619 VL - 50 IS - 6 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.791344 UR - https://doi.org/10.15672/hujms.791344 Y2 - 2021 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Disjoint and simultaneous hypercyclic Rolewicz-type operators %A Nurhan Çolakoğlu , Özgür Martin %T Disjoint and simultaneous hypercyclic Rolewicz-type operators %D 2021 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 50 %N 6 %R doi: 10.15672/hujms.791344 %U 10.15672/hujms.791344
ISNAD Çolakoğlu, Nurhan , Martin, Özgür . "Disjoint and simultaneous hypercyclic Rolewicz-type operators". Hacettepe Journal of Mathematics and Statistics 50 / 6 (December 2021): 1609-1619 . https://doi.org/10.15672/hujms.791344
AMA Çolakoğlu N. , Martin Ö. Disjoint and simultaneous hypercyclic Rolewicz-type operators. Hacettepe Journal of Mathematics and Statistics. 2021; 50(6): 1609-1619.
Vancouver Çolakoğlu N. , Martin Ö. Disjoint and simultaneous hypercyclic Rolewicz-type operators. Hacettepe Journal of Mathematics and Statistics. 2021; 50(6): 1609-1619.
IEEE N. Çolakoğlu and Ö. Martin , "Disjoint and simultaneous hypercyclic Rolewicz-type operators", Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 6, pp. 1609-1619, Dec. 2021, doi:10.15672/hujms.791344