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Notes on the \({\sin 2 \Theta}\) Theorem

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Abstract

An analogue of the Davis-Kahan \({\sin 2\Theta}\) theorem from (SIAM J Numer Anal 7:1–46, 1970) is proved under a general spectral separation condition. This extends the generic \({\sin 2\theta}\) estimates recently shown by Albeverio and Motovilov in (Complex Anal Oper Theory 7:1389–1416, 2013). The result is applied to the subspace perturbation problem to obtain a bound on the arcsine of the norm of the difference of the spectral projections associated with isolated components of the spectrum of the unperturbed and perturbed operators, respectively.

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References

  1. Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications, New York (1993)

    MATH  Google Scholar 

  2. Albeverio S., Makarov K.A., Motovilov A.K.: Graph subspaces and the spectral shift function. Can. J. Math. 55, 449–503 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Albeverio S., Motovilov A.K.: Operator Stieltjes integrals with respect to a spectral measure and solutions to some operator equations. Trans. Moscow Math. Soc. 72, 45–77 (2011)

    Article  MathSciNet  Google Scholar 

  4. Albeverio S., Motovilov A.K.: The a priori \({\tan\Theta}\) theorem for spectral subspaces. Integral Equ. Oper. Theory 73, 413–430 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Albeverio S., Motovilov A.K.: Sharpening the norm bound in the subspace perturbation theory. Complex Anal. Oper. Theory 7, 1389–1416 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Alizadeh R., Asadi M.: An extension of Ky Fan’s dominance theorem. Banach J. Math. Anal. 6, 139–146 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bhatia R., Davis C., Koosis P.: An extremal problem in Fourier analysis with applications to operator theory. J. Funct. Anal. 82, 138–150 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bhatia R., Davis C., McIntosh A.: Perturbation of spectral subspaces and solution of linear operator equations. Linear Algebra Appl. 52/53, 45–67 (1983)

    Article  MathSciNet  Google Scholar 

  9. Bhatia R., Rosenthal P.: How and why to solve the operator equation AXXB = Y. Bull. Lond. Math. Soc. 29, 1–21 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Davis C.: Separation of two linear subspaces. Acta Sci. Math. Szeged 19, 172–187 (1958)

    MATH  MathSciNet  Google Scholar 

  11. Davis C.: The rotation of eigenvectors by a perturbation. J. Math. Anal. Appl. 6, 159–173 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  12. Davis C., Kahan W.M.: The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal. 7, 1–46 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gohberg, I. C., Kreĭn, M. G.: Introduction to the Theory of Linear Nonselfad-joint Operators. Translations of Mathematical Monographs, vol. 18, Am. Math. Soc., Providence, RI (1969)

  14. Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  MATH  Google Scholar 

  15. Kostrykin V., Makarov K.A., Motovilov A.K.: On a subspace perturbation problem. Proc. Am. Math. Soc. 131, 3469–3476 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kostrykin, V., Makarov, K. A., Motovilov, A.K.: Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach. In: Advances in Differential Equations and Mathematical Physics (Birmingham, 2002), Contemp. Math., vol. 327, pp. 181–198. Am. Math. Soc., Providence, RI (2003)

  17. Kostrykin, V., Makarov, K.A., Motovilov, A. K.: A generalization of the \({\tan2\Theta}\) theorem. In: Current Trends in Operator Theory and Its Applications. Operator Theory: Advances and Applications, vol. 149, pp. 349–372. Birkhäuser, Basel (2004)

  18. Kostrykin V., Makarov K.A., Motovilov A.K.: On the existence of solutions to the operator Riccati equation and the \({\tan\Theta}\) theorem. Integral Equ. Oper. Theory 51, 121–140 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kostrykin V., Makarov K.A., Motovilov A.K.: Perturbation of spectra and spectral subspaces. Trans. Am. Math. Soc. 359, 77–89 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. McEachin R.: A sharp estimate in an operator inequality. Proc. Am. Math. Soc. 115, 161–165 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. McEachin R.: Closing the gap in a subspace perturbation bound. Linear Algebra Appl. 180, 7–15 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. McEachin R.: Analyzing specific cases of an operator inequality. Linear Algebra Appl. 208/209, 343–365 (1994)

    Article  MathSciNet  Google Scholar 

  23. Makarov, K. A., Schmitz, S., Seelmann, A.: Reducing graph subspaces and strong solutions to operator Riccati equations. e-print arXiv:1307.6439 [math.SP] (2013)

  24. Motovilov A.K., Selin A.V.: Some sharp norm estimates in the subspace perturbation problem. Integral Equ. Oper. Theory 56, 511–542 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Seelmann, A.: On an estimate in the subspace perturbation problem. e-print arXiv:1310.4360 [math.SP] (2013)

  26. Sz.-Nagy B.: Über die Ungleichnung von H. Bohr. Math. Nachr. 9, 255–259 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sz.-Nagy, B.: Bohr inequality and an operator equation. In: Operators in Indefinite Metric Spaces, Scattering Theory and Other Topics (Bucharest, 1985). Operator Theory: Advances and Applications, vol. 24, pp. 321–327. Birkhäuser, Basel (1987)

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Correspondence to Albrecht Seelmann.

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The material presented in this work will be part of the author’s Ph.D. thesis.

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Seelmann, A. Notes on the \({\sin 2 \Theta}\) Theorem. Integr. Equ. Oper. Theory 79, 579–597 (2014). https://doi.org/10.1007/s00020-014-2127-z

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  • DOI: https://doi.org/10.1007/s00020-014-2127-z

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