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Variation of the discrete eigenvalues of normal operators. (English) Zbl 0831.47016

Summary: The Hoffman-Wielandt inequality, which gives a bound for the distance between the spectra of two normal matrices, is generalized to normal operators \(A\), \(B\) on a separable Hilbert space, such that \(A- B\) is Hilbert-Schmidt.

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A55 Perturbation theory of linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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References:

[1] I. David Berg, An extension of the Weyl-von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365 – 371. · Zbl 0212.15903
[2] Rajendra Bhatia and Tirthankar Bhattacharyya, A generalization of the Hoffman-Wielandt theorem, Linear Algebra Appl. 179 (1993), 11 – 17. · Zbl 0774.15011 · doi:10.1016/0024-3795(93)90318-I
[3] Rajendra Bhatia and Ludwig Elsner, The Hoffman-Wielandt inequality in infinite dimensions, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 3, 483 – 494. · Zbl 0805.47017 · doi:10.1007/BF02867116
[4] Rajendra Bhatia and Kalyan B. Sinha, A unitary analogue of Kato’s theorem on variation of discrete spectra, Lett. Math. Phys. 15 (1988), no. 3, 201 – 204. · Zbl 0658.47008 · doi:10.1007/BF00398588
[5] James Alan Cochran and Erold W. Hinds, Improved error bounds for the eigenvalues of certain normal operators, SIAM J. Numer. Anal. 9 (1972), 446 – 453. · Zbl 0256.47009 · doi:10.1137/0709040
[6] L. Elsner, A note on the Hoffman-Wielandt theorem, Linear Algebra Appl. 182 (1993), 235 – 237. · Zbl 0803.15023 · doi:10.1016/0024-3795(93)90501-E
[7] Shmuel Friedland, Inverse eigenvalue problems, Linear Algebra and Appl. 17 (1977), no. 1, 15 – 51. · Zbl 0358.15007
[8] I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. · Zbl 0181.13503
[9] A. J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1953), 37 – 39. · Zbl 0051.00903
[10] Tosio Kato, Variation of discrete spectra, Comm. Math. Phys. 111 (1987), no. 3, 501 – 504. · Zbl 0632.47002
[11] John von Neumann, Collected works. Vol. IV: Continuous geometry and other topics, General editor: A. H. Taub, Pergamon Press, Oxford-London-New York-Paris, 1962. · Zbl 0188.00103
[12] Dan Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory 2 (1979), no. 1, 3 – 37. · Zbl 0446.47003
[13] H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo 27 (1909), 373-392. · JFM 40.0395.01
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