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Spectral continuity and the dynamics of quasi-2-isometric operators

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Abstract

Linear dynamics has been a rapidly evolving area since the early 1990s. It lies at the intersection of operator theory and topological dynamics and its central property is hypercyclycity. The class of quasi-2-isometric operators on Hilbert space extends the classes of 2-isometric operators due to Agler and Stankus and quasi-isometries by S. M. Patel. An operator T on a complex Hilbert space is called quasi-2-isometry if

$$\begin{aligned} T^{*3}T^{3 }- 2T^{*2 }T^{2}+ T^{*}T = 0. \end{aligned}$$

In the present article, we prove that a weakly supercyclic quasi-2-isometric operator is a unitary operator and a quasi-2-isometric operator is not weakly hypercyclic. We also, show that the spectrum is continuous on the class of all quasi-2-isometric operators and Weyl’s theorem holds for quasi-2-isometric operators.

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References

  1. Aiena, P. 2004. Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer Academic Publishers.

  2. Agler, J., and M. Stankus. 1995. m-isometric transformations of Hilbert space I. Integral Equations and Operator Theory 21: 387–429.

    Article  MathSciNet  Google Scholar 

  3. Agler, J., and M. Stankus. 1995. m-isometric transformations of Hilbert space II. Integral Equations and Operator Theory 21: 1–48.

    Article  MathSciNet  Google Scholar 

  4. Agler, J., and M. Stankus. 1995. m-isometric transformations of Hilbert space III. Integral Equations and Operator Theory 21: 379–421.

    Article  MathSciNet  Google Scholar 

  5. Athavale, A. 1991. Some operator theoretic calculus for positive definite kernels. Proceedings of the American Mathematical Society 112: 701–708.

    MathSciNet  Google Scholar 

  6. Aiena, P. 2004. Fredhlom and Local Spectral Theory with Applications to Multipliers. Dordrecht: Kluwer Acad. Publishers.

    Google Scholar 

  7. Aiena, P., J.R. Guillen, and P. Pena. 2013. Unifying approach to Weyl type theorems for Banach Space operators. Integral Equations and Operator Theory 77: 371–384.

    Article  MathSciNet  Google Scholar 

  8. Bermudez, T., A. Martinon, and J.A. Noda. 2013. Products of m-isometries. Linear Algebra and its Applications 438: 80–86.

    Article  MathSciNet  Google Scholar 

  9. Bayart, F., and E. Matheron. 2006. Hyponormal operators, weighted shifts and weak forms of supercyclicity. Proceedings of the Edinberg Mathematical Society 49: 1–15.

    Article  MathSciNet  Google Scholar 

  10. Botelho, F., and J. Jamison. 2010. Isometric properties of elementary operators. Linear Algebra and its Applications 432: 357–365.

    Article  MathSciNet  Google Scholar 

  11. Bermudez, T., I. Marrero, and A. Martinon. 2009. On the orbit of an m-isometry. Integral Equations and Operator Theory 64: 487–494.

    Article  MathSciNet  Google Scholar 

  12. Bourdon, P.S. 1997. Orbits of hyponormal operators. Michigan Mathematical Journal 44: 345–353.

    Article  MathSciNet  Google Scholar 

  13. Berberian, S.K. 1962. Approximate proper vectors. Proceedings of the American Mathematical Society 13: 111–114.

    Article  MathSciNet  Google Scholar 

  14. Cho, M., S. Ota, K. Tanahashi, and A. Uchiyama. 2012. Spectral properties of \(m\)-isometric operators. Functional analysis, Approximation and Computation 42: 33–39.

    MathSciNet  Google Scholar 

  15. Cho, M., S. Ota, and K. Tanahashi. 2013. Invertible weighted shift operators which are m-isometries. Proceedings of the American Mathematical Society 141: 4241–4247.

    Article  MathSciNet  Google Scholar 

  16. Conway, J.B., and B.B. Morrel. 1979. Operators that are points of spectral continuity. Integral Equations Operator Theory 2: 174–198.

    Article  MathSciNet  Google Scholar 

  17. Duggal, B.P., I.H. Jeon, and I.H. Kim. 2010. Continuity of the spectrum on a class of upper triangular operator matrices. Journal of Mathematical Analysis and Applications 370: 584–587.

    Article  MathSciNet  Google Scholar 

  18. Feldman, N.S. 2002. n-supercyclic operators. Studia Mathematica 151: 141–159.

    Article  MathSciNet  Google Scholar 

  19. Gleason, J., and S. Richter. 2006. \(m\)-isometries commuting \(n\)-tuple of operators on a Hilbert space. Integral Equations and Operator Theory 56: 181–196.

    Article  MathSciNet  Google Scholar 

  20. Helling, C. 2008. Two isometries on Pontryagin space. Integral Equations and Operator Theory 61: 211–239.

    Article  MathSciNet  Google Scholar 

  21. In Ho Jean and In Hyoun Kim. 2006. On operators satisfying \(T^{*}|T^{2}|T=T^{*} |T|^{2}T\). Linear Algebra and Its Applications 418: 854–862.

    MathSciNet  Google Scholar 

  22. Halmos, P.R. 1967. A Hilbert Space Problem Book. Princeton: Van Nostrand.

    Google Scholar 

  23. Harte, R.E., and W.Y. Lee. 1997. Another note on Weyl theorem. Transactions of the American Mathematical Society 349: 2115–2124.

    Article  MathSciNet  Google Scholar 

  24. Jung, S., and J.E. Lee. 2012. On (\(A, m\))-expansive operators. Studia Mathematica 213: 3–23.

    Article  MathSciNet  Google Scholar 

  25. Mecheri, S., and S. M. Patel, 2017. On quasi-2-isometric operators. Linear Multi-Linear Algebra, 1019–1025. https://doi.org/10.1080/03081087.2017.1335283.

  26. Mecheri, S., and T. Prasad, 2016. On \(n\)-quasi-\(m\)-isometric operators. Asian European Journal of Mathematics 9 (4): 1650073.

  27. Mecheri, S. 2012. Bishop’s property, SVEP and Dunford property (C). The Electronic Journal of Linear Algebra 23: 523–529.

    Article  MathSciNet  Google Scholar 

  28. Mecheri, S. 2013. On k-quasi-M-hyponormal operators. Mathematical Inequalities and Applications 16: 895–902.

    Article  MathSciNet  Google Scholar 

  29. Mecheri, S., and S. Makhlouf. 2008. Weyl type theorems for posinormal operators. Mathematical Proceedings of the Royal Irish Academy 108A: 69–79.

    Article  MathSciNet  Google Scholar 

  30. Patel, S.M. 2000. A note on quasi-isometries. Glasnik Matematicki 35 (55): 307–312.

    MathSciNet  Google Scholar 

  31. Patel, S.M. 2002. 2-isometric operators. Glasnik Matematicki 37 (57): 143–147.

    MathSciNet  Google Scholar 

  32. Richter, S. 1991. A representation theorem for Cyclic Analytic 2-isometries. Transactions of the American Mathematical Society 238: 325–349.

    Article  MathSciNet  Google Scholar 

  33. Richter, S. 1988. Invariant subspaces of the Dirichlet shift. Journal fur die reine und angewandte Mathematik 386 (386): 205–220.

    MathSciNet  Google Scholar 

  34. Shimorin, S.M. 2001. World type decompositions and wandering subspaces close to isometries. Journal fur die reine und angewandte Mathematik 531: 147–189.

    MathSciNet  Google Scholar 

  35. Shimorin, S.M. 2001. Wold-type decompositions and wandering subspaces for operators close to isometries. Journal fur die reine und Angewandte Mathematik 531: 147–189.

    MathSciNet  Google Scholar 

  36. Sanders, R. 2004. Weakly supercyclic operators. Journal of Mathematical Analysis and Applications 292: 148–159.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We wish to thank the referee for careful reading of the paper and valuable comments for the origin version of the work.

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Authors and Affiliations

  1. College of Mathematics and Informatics, Department of Mathematics, Mohamed El Bachir El Ibrahimi University, Bourdj Bou Arreridj, Algeria

    Salah Mecheri

  2. Laboratory of Mathematics and Applications LR17ES11, Faculty of Sciences of Gabes, University of Gabes, Gabes, Tunisia

    Adel Saddi, Amira Sedki & Samir Sbita

  3. Department of Mathematics, University of Gabes, Gabes, Tunisia

    Adel Saddi, Amira Sedki & Samir Sbita

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  1. Salah Mecheri
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Correspondence to Salah Mecheri.

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Communicated by Gadadhar Misra.

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Mecheri, S., Saddi, A., Sedki, A. et al. Spectral continuity and the dynamics of quasi-2-isometric operators. J Anal (2024). https://doi.org/10.1007/s41478-024-00758-9

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  • DOI: https://doi.org/10.1007/s41478-024-00758-9

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