The Hoffman-Wielandt inequality in infinite dimensions

  • Rajendra Bhatia
  • Ludwig Elsner


The Hoffman-Wielandt inequality for the distance between the eigenvalues of two normal matrices is extended to Hilbert-Schmidt operators. Analogues for other norms are obtained in a special case.


Hoffman-Wielandt inequality infinite dimensions Hilbert-Schmidt operators Schattenp-norms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ando T and Bhatia R, Eigenvalue inequalities associated with the Cartesian decomposition,Linear Multilinear Algebra,22 (1987) 133–147MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Bhatia R, Analysis of spectral variation and some inequalities,Trans. Am. Math. Soc.,272 (1982) 323–331MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Bhatia R,Perturbation Bounds for Matrix Eigenvalues (Essex Longman) (1987)MATHGoogle Scholar
  4. [4]
    Bhatia R and Bhattacharyya T, A generalisation of the Hoffman-Wielandt theorem,Linear Algebra Appl.,179 (1993) 11–17MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Bhatia R and Davis C, A bound for the spectral variation of a unitary operator,Linear Multilinear Algebra,15 (1984) 71–76MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Bhatia R, Davis C and Koosis P, An extremal problem in Fourier analysis with applications to operator theory,J. Funct. Anal.,82 (1989) 138–150MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Bhatia R, Davis C and McIntosh A, Perturbation of spectral subspaces and solution of linear operator equations,Linear Algebra Appl. 52 (1983) 45–67MathSciNetGoogle Scholar
  8. [8]
    Bhatia R and Holbrook J A R, Short normal paths and spectral variation,Proc. Am. Math. Soc.,94 (1985) 377–382MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Bhatia R and Holbrook J A R, On the Clarkson-McCarthy inequalities,Math. Ann. 281 (1988) 7–12MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Bhatia R and Sinha K B, A unitary analogue of Kato's theorem on variation of discrete spectra,Lett. Math. Phys.,15 (1988) 201–204MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Elsner L, A note on the Hoffman-Wielandt theorem,Linear Algebra Appl.,182 (1993) 235–237MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Friedland S, Inverse eigenvalue problems,Linear Algebra Appl.,17 (1977) 15–51MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Hoffman A J and Wielandt H W, The variation of the spectrum of a normal matrix,Duke Math. J.,20 (1953) 37–39MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Kato T, Variation of discrete spectra,Commun. Math. Phys.,111 (1987) 501–504MATHCrossRefGoogle Scholar
  15. [15]
    Lidskii V B, The proper values of the sum and product of symmetric matrices,Dokl. Akad. Nauk, SSSR,75 (1950) 769–772MathSciNetGoogle Scholar
  16. [16]
    Markus A S, The eigen and singular values of the sum and product of linear operators,Russ. Math, Surv.,19 (1964) 92–120CrossRefMathSciNetGoogle Scholar
  17. [17]
    McIntosh A, Pryde A and Ricker W, Comparison of joint spectra for certain classes of commuting operators,Stud. Math.,88 (1988) 23–36MATHMathSciNetGoogle Scholar
  18. [18]
    Sakai Y, Continuous versions of an inequality due to Hoffman and Wielandt,Linear Algebra Appl.,71 (1985) 283–287MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Simon B,Trace Ideals and Their Applications, (Cambridge: University Press), (1979)MATHGoogle Scholar
  20. [20]
    Sunder V S, Distance between normal operators,Proc. Am. Math. Soc.,84 (1982) 483–484MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 1994

Authors and Affiliations

  • Rajendra Bhatia
    • 1
  • Ludwig Elsner
    • 2
  1. 1.Indian Statistical InstituteNew DelhiIndia
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany

Personalised recommendations