Subspace-chaotic operators and their direct sum

Document Type : Research Paper

Authors

1 Department of Mathematics, Farhangian University, Tarbiat Moallem St., Tehran, Iran

2 Department of Mathematics, University of Maragheh, University Boulevard, Maragheh, Iran

Abstract

In this article, we state some sufficient conditions for an operator to be subspace-chaotic. The properties of the direct sum of operators are interesting topics in operator theory. In this paper, we show that the direct sum of the two subspace-chaotic operators is subspace-chaotic with respect to at least two subspaces, and the same is true for the subspace-transitive and subspace-hypercyclic operators. We also make interesting examples of these operators with the help of the stated theorems. In addition, we prove that if the direct sum of two operators is subspace-chaotic or subspace-transitive, then each of those two operators is subspace-chaotic or subspace-transitive, respectively. We also generalize these results to the direct sum of a finite number of these operators.

Keywords

References

[1] N. Bamerni and A. Kilicman, On the direct sum of two bounded linear operators and subspace-hypercyclicity, General Letters in Mathematics, 8 (2020) 1–7.
[2] N. Bamerni, V. Kadets and A. Kilicman, Hypercyclic operators are subspace hypercyclic, J. Math. Anal. Appl., 435 (2016) 1812–1815.
[3] F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge University Press, Cambridge, 2009.
[4] K. G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Springer-Verlag London Ltd., London, 2011.
[5] D. A. Herrero, Hypercyclic operators and chaos, J. Oper. Theory, 28 (1992) 93–103.
[6] R. R. Jimenez-Munguila, R. A. Martinez-Avendano and A. Peris, Some questions about subspace-hypercyclic operators, J. Math. Anal. Appl., 408 (2013) 209–212.
[7] C. M. Le, On subspace-hypercyclic operators, Proc. Am. Math. Soc., 139 (2011) 2847–2852.
[8] B. F. Madore and R. A. Martinez-Avendano, Subspace hypercyclicity, J. Math. Anal. Appl., 373 (2011) 502–511.
[9] R. A. Martinez-Avendano and O. Zatarain-Vera, Subspace-hypercyclicity for Toeplitz operators, J. Math. Anal. Appl., 422 (2015) 772–775.
[10] M. Moosapoor, On multi subspace-hypercyclic operators,Commun. Korean Math. Soc., 35 (2020) 1185–1192.
[11] S. Talebi and M. Moosapoor, Subspace-chaotic and subspace-weakly mixing operators, Int. J. Pure Appl. Math., 78 (2012) 879–885.
[12] L. Zhang and Z. H. Zhou, Notes about subspace-supercyclic operators, Ann. Funct. Anal., 6 (2015) 60–68.

Files

History

  • Receive Date: 27 November 2021
  • Revise Date: 05 February 2022
  • Accept Date: 07 February 2022
  • Publish Date: 23 August 2021

Share

How to cite

Statistics