DOI QR코드

DOI QR Code

DYNAMICAL SYSTEMS AND λ-ALUTHGE TRANSFORMS

  • Tran, Linh (Department of Mathematics Chungnam National University)
  • Received : 2022.05.17
  • Accepted : 2022.08.24
  • Published : 2022.08.15

Abstract

In this note, we verify that the bounded shadowing property and quasi-hyperbolicity of bounded linear operators on Hilbert spaces are preserved under Aluthge transforms.

Keywords

References

  1. J. Antezana, E. R. Pujals and D. Stojanoff, The iterated Aluthge transforms of matrix converge, Adv. Math. (N Y), 226 (2011), 1591-1620. https://doi.org/10.1016/j.aim.2010.08.012
  2. A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equ. Oper. Theory, 13 (1990), 307-315. https://doi.org/10.1007/BF01199886
  3. C. K. K. Batty and Y. Tomilov, Quasi-hyperbolic semigroups, J. Funct. Anal., 258 (2010), 3855-3878. https://doi.org/10.1016/j.jfa.2010.01.005
  4. N. C. Bernades Jr, P. R. Cirilo, U. B. Darji, A. Messaoudi and E. R. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl., 461 (2018), 796-816. https://doi.org/10.1016/j.jmaa.2017.11.059
  5. N. C. Bernades Jr and A. Messaoudi, Shadowing and Structral Stability in Linear Dynamical Systems, Ergod. Theory Dyn. Syst., 41 (2021), no. 4, 961-980. https://doi.org/10.1017/etds.2019.107
  6. P. Cirilo, B. Gollobit and E. Pujals, Dynamics of generalized hyperbolic operators, Adv. Math. (N Y), 387 (2021), 107830. https://doi.org/10.1016/j.aim.2021.107830
  7. E. D'Aniello, U.B. Darji, M. Maiuriello, Generalized Hyperbolicity and Shadowing in Lp spaces, J. Differ. Equ., 298 (2021), 68-94. https://doi.org/10.1016/j.jde.2021.06.038
  8. T. Huruya, A note on p-hyponormal operators, Proc. Am. Math. Soc., 125 (1997), 3617-3624. https://doi.org/10.1090/S0002-9939-97-04004-5
  9. R. Harte, Y. O. Kim and W. Y. Lee, Spectral pictures of AB and BA, Proc. Am. Math. Soc., 134 (2005), no. 1 ,105-110. https://doi.org/10.1090/S0002-9939-05-08015-9
  10. I.B. Jung, E. Ko and C. Pearcy, Aluthge tranforms of operators, Integral Equ. Oper. Theory, 37 (2000), 437-448. https://doi.org/10.1007/BF01192831
  11. I.B. Jung, E. Ko and C. Pearcy,Aluthge transforms of operators, Trends Math., Information Center for Mathematical Sciences, 6 (2003) no. 2, 59-68.
  12. S. H . Lee, W. Y. Lee and J. Yoon, The mean transform of bounded linear operators, J. Math. Anal. Appl., 410 (2014), 70-81. https://doi.org/10.1016/j.jmaa.2013.08.003
  13. K. Lee, C. A. Morales, Hyperbolicity, shadowing, and bounded orbits, Qual. Theory Dyn. Syst., 21 (2022), no. 61, DOI: https://doi.org/10.1007/s12346-022-00588-9.
  14. P. Y. Wu, Numerical range of Aluthge transform of operator, Linear Algebra Appl., 357 (2002), 295-298. https://doi.org/10.1016/S0024-3795(02)00361-0
  15. T. Yamazaki, An expression of spectral radius via Aluthge transformation, Proc Am Math Soc., 130 (2002), 1131-1137. https://doi.org/10.1090/S0002-9939-01-06283-9
  16. T. Yamazaki, On numerical range of the Aluthge transformation , Linear Algebra Appl., 341 (2002), 111-117. https://doi.org/10.1016/S0024-3795(01)00333-0
  17. T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Stud. Math., 178 (2007), 83-89. https://doi.org/10.4064/sm178-1-5