Opuscula Math. 44, no. 2 (2024), 241-248
https://doi.org/10.7494/OpMath.2024.44.2.241

 
Opuscula Mathematica

An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components

Michael Gil'

Abstract. Let \(A\) be a bounded linear operator in a complex separable Hilbert space, \(A^*\) be its adjoint one and \(A_I:=(A-A^*)/(2i)\). Assuming that \(A_I\) is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of \(A\). Our results are formulated in terms of the "extended" eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality \(\sum_{k=1}^\infty (\operatorname{Im} \lambda_k(A))^2 \leq N_2^2(A_I)\), where \(\lambda_k(A)\) \((k=1,2, \ldots )\) are the eigenvalues of \(A\) and \(N_2(\cdot)\) is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.

Keywords: Hilbert space, linear operators, eigenvalues.

Mathematics Subject Classification: 47A10, 47A55, 47B10.

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  1. P. Aiena, S. Triolo, Weyl-type theorems on Banach spaces under compact perturbations, Mediterr. J. Math. 15 (2018), no. 3, Article no. 126. https://doi.org/10.1007/s00009-018-1176-y
  2. R. Bhatia, L. Elsner, The Hoffman-Wielandt inequality in infinite dimensions, Proc. Indian Acad. Sci. (Math. Sci.) 104 (1994), no. 3, 483-494. https://doi.org/10.1007/BF02867116
  3. V.S. Budyka, M.M. Malamud, Deficiency indices and discreteness property of block Jacobi matrices and Dirac operators with point interactions, J. Math. Anal. Appl. 506 (2022), no. 1, Article no. 125582. https://doi.org/10.1016/j.jmaa.2021.125582
  4. W. Chaker, A. Jeribi, B. Krichen, Demicompact linear operators, essential spectrum and some perturbation results, Math. Nachr. 288 (2015), no. 13, 1476-1486. https://doi.org/10.1002/mana.201200007
  5. W. Geng, K. Tao, Large deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-Rüssmann frequency, Commun. Pure Appl. Anal. 19 (2020), no. 12, 5305-5335. https://doi.org/10.3934/cpaa.2020240
  6. M.I. Gil', Lower bounds for eigenvalues of Schatten-von Neumann operators, J. Inequal. Pure Appl. Math. 8 (2007), no. 3, Article no. 66.
  7. M.I. Gil', Sums of real parts of eigenvalues of perturbed matrices, J. Math. Inequal. 4 (2010), no. 4, 517-522. https://doi.org/10.7153/jmi-04-46
  8. M.I. Gil', Bounds for eigenvalues of Schatten-von Neumann operators via self-commutators, J. Funct. Anal. 267 (2014), no. 9, 3500-3506. https://doi.org/10.1016/j.jfa.2014.06.019
  9. M.I. Gil', A bound for imaginary parts of eigenvalues of Hilbert-Schmidt operators, Funct. Anal. Approx. Comput. 7 (2015), no. 1, 35-38.
  10. M.I. Gil', Inequalities for eigenvalues of compact operators in a Hilbert space, Commun. Contemp. Math. 18 (2016), no. 1, Article no. 1550022. https://doi.org/10.1142/S0219199715500224
  11. M.I. Gil', Operator Functions and Operator Equations, World Scientific, New Jersey, 2018. https://doi.org/10.1142/10482
  12. M.I. Gil', Norm estimates for resolvents of linear operators in a Banach space and spectral variations, Adv. Oper. Theory 4 (2019), no. 1, 113-139. https://doi.org/10.15352/aot.1801-1293
  13. M.I. Gil', Bounds for absolute values and imaginary parts of matrix eigenvalues via traces, Proyecciones 41 (2022), no. 5, 1229-1237. https://doi.org/10.22199/issn.0717-6279-5349
  14. M.I. Gil', On matching distance between eigenvalues of unbounded operators, Constr. Math. Anal.5 (2022), no. 1, 46-53. https://doi.org/10.33205/cma.1060718
  15. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Mathem. Monographs, vol. 18, Amer. Math. Soc., Providence, R. I., 1969.
  16. I.H. Gumus, O. Hirzallah, F. Kittaneh, Eigenvalue localization for complex matrices, Electron. J. Linear Algebra 27 (2014), 892-906. https://doi.org/10.13001/1081-3810.2866
  17. A. Jeribi, Perturbation Theory for Linear Operators. Denseness and Bases with Applications, Springer-Verlag Singapore, 2021. https://doi.org/10.1007/978-981-16-2528-2
  18. W. Kahan, Spectra of nearly Hermitian matrices, Proc. Amer. Math. Soc. 48 (1975), 11-17. https://doi.org/10.2307/2040683
  19. T. Kato, Variation of discrete spectra, Commun. Math. Phys. 111 (1987), 501-504. https://doi.org/10.1007/BF01238911
  20. M. Kian, M. Bakherad, A new estimation for eigenvalues of matrix power functions, Anal. Math. 45 (2019), no. 3, 527-534. https://doi.org/10.1007/s10476-019-0912-2
  21. M. Malejki, Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices, Opuscula Math. 27 (2007), no. 1, 37-49.
  22. M. Malejki, Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in \(l^2\) by the use of finite submatrices, Cent. Eur. J. Math. 8 (2010), 114-128. https://doi.org/10.2478/s11533-009-0064-x
  23. O. Rojo, Inequalities involving the mean and the standard deviation of nonnegative real numbers, J. Inequal. Appl. (2006), Article no. 43465, 1-15. https://doi.org/10.1155/JIA/2006/43465
  24. O. Rojo, R.L. Soto, H. Rojo, New eigenvalue estimates for complex matrices, Comput. Math. Appl. 25 (1993), no. 3, 91-97. https://doi.org/10.1016/0898-1221(93)90147-N
  25. M.L. Sahari, A.K. Taha, L. Randriamihamison, A note on the spectrum of diagonal perturbation of weighted shift operator, Matematiche (Catania) 74 (2019), no. 1, 35-47. https://doi.org/10.4418/2019.74.1.3
  26. M. Webb, S. Olver, Spectra of Jacobi operators via connection coefficient matrices, Comm. Math. Phys. 382 (2021), no. 2, 657-707. https://doi.org/10.1007/s00220-021-03939-w
  • Michael Gil'
  • Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
  • Communicated by P.A. Cojuhari.
  • Received: 2023-07-02.
  • Revised: 2023-12-03.
  • Accepted: 2023-12-03.
  • Published online: 2024-01-15.
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Cite this article as:
Michael Gil', An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components, Opuscula Math. 44, no. 2 (2024), 241-248, https://doi.org/10.7494/OpMath.2024.44.2.241

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