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Spectral radius of semi-Hilbertian space operators and its applications

Abstract

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space \(\mathcal {H}\), which are bounded with respect to the seminorm induced by a positive operator A on \(\mathcal {H}\). Mainly, we show that \(r_A(T)\le \omega _A(T)\) for every A-bounded operator T, where \(r_A(T)\) and \(\omega _A(T)\) denote respectively the A-spectral radius and the A-numerical radius of T. This allows to establish that \(r_A(T)=\omega _A(T)=\Vert T\Vert _A\) for every A-normaloid operator T, where \(\Vert T\Vert _A\) is denoted to be the A-operator seminorm of T. Moreover, some characterizations of A-normaloid and A-spectraloid operators are given.

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References

  1. 1.

    Arias, M.L., Corach, G., Gonzalez, M.C.: Partial isometries in semi-Hilbertian spaces. Linear Algebra Appl. 428(7), 1460–1475 (2008)

  2. 2.

    Arias, M.L., Corach, G., Gonzalez, M.C.: Metric properties of projections in semi-Hilbertian spaces. Integr. Equ. Oper. Theory 62, 11–28 (2008)

  3. 3.

    Arias, M.L., Corach, G., Gonzalez, M.C.: Lifting properties in operator ranges. Acta Sci. Math. (Szeged) 75(3–4), 635–653 (2009)

  4. 4.

    Baklouti, H., Feki, K.: On joint spectral radius of commuting operators in Hilbert spaces. Linear Algebra Appl. 557, 455–463 (2018)

  5. 5.

    Baklouti, H., Feki, K., Sid Ahmed, O.A.M.: Joint numerical ranges of operators in semi-Hilbertian spaces. Linear Algebra Appl. 555, 266–284 (2018)

  6. 6.

    Baklouti, H., Feki, K., Sid Ahmed, O.A.M.: Joint normality of operators in semi-Hilbertian spaces. Linear Multilinear Algebra (2019). https://doi.org/10.1080/03081087.2019.1593925

  7. 7.

    Berger, C.: A strange dilation theorem, Abstract 625–152. Not. Am. Math. Soc. 12, 590 (1965)

  8. 8.

    Chan, J.-T., Chan, K.: An observation about normaloid operators. Oper. Matrices 11(3), 885–890 (2017)

  9. 9.

    de Branges, L., Rovnyak, J.: Square Summable Power Series. Holt, Rinehert and Winston, New York (1966)

  10. 10.

    Douglas, R.G.: On majorization, factorization and range inclusion of operators in Hilbert space. Proc. Am. Math. Soc. 17, 413–416 (1966)

  11. 11.

    Faghih-Ahmadi, M., Gorjizadeh, F.: A-numerical radius of A-normal operators in semi-Hilbertian spaces. Ital. J. Pure Appl. Math. 36, 73–78 (2016)

  12. 12.

    Fekete, M.: Über der Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17, 228–249 (1923)

  13. 13.

    Furuta, T., Takeda, Z.: A characterization of spectraloid operators and its generalization Proc. Jpn. Acad. 43(7), 599–604 (1967)

  14. 14.

    Furuta, T.: On the class of paranormal operators proc. Jpn. Acad. 43(7), 594–598 (1967)

  15. 15.

    Goldberg, M., Tadmor, E.: On the numerical radius and its applications. Linear Algebra Appl. 42, 263–284 (1982)

  16. 16.

    Kittaneh, F.: A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Stud. Math. 158, 11–17 (2003)

  17. 17.

    Kubrusly, C.S.: The Elements of Operator Theory, 2nd edn. Springer, New York (2011)

  18. 18.

    Majdak, W., Secelean, N.A., Suciu, L.: Ergodic properties of operators in some semi-Hilbertian spaces. Linear Multilinear Algebra 61(2), 139–159 (2013)

  19. 19.

    Saddi, A.: $A$-Normal operators in semi-Hilbertian spaces. Aust. J. Math. Anal. Appl. 9(1), 1–12 (2012)

  20. 20.

    Sid Ahmed, O.A.M., Benali, A.: Hyponormal and k-quasi-hyponormal operators on semi-Hilbertian spaces. Aust. J. Math. Anal. Appl. 13(1), 1–22 (2016)

  21. 21.

    Zamani, A.: $A$-numerical radius inequalities for semi-Hilbertian space operators. Linear Algebra Appl. 578, 159–183 (2019)

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Author information

Correspondence to Kais Feki.

Additional information

Communicated by Takeaki Yamazaki.

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Feki, K. Spectral radius of semi-Hilbertian space operators and its applications. Ann. Funct. Anal. (2020). https://doi.org/10.1007/s43034-020-00064-y

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Keywords

  • Positive operator
  • Semi-inner product
  • Spectral radius
  • Numerical radius
  • Normaloid operator
  • Spectraloid operator

Mathematics Subject Classification

  • 46C05
  • 47A12
  • 47B65
  • 47B15
  • 47B20