Perturbation of spectra and spectral subspaces

Kostrykin, Vadim; Makarov, K.; Motovilov, A.

doi:10.1090/S0002-9947-06-03930-4

Perturbation of spectra and spectral subspaces
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by Vadim Kostrykin, K. A. Makarov and A. K. Motovilov PDF

Trans. Amer. Math. Soc. 359 (2007), 77-89

Abstract:

We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively.

References

N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications, Inc., New York, 1993. Translated from the Russian and with a preface by Merlynd Nestell; Reprint of the 1961 and 1963 translations; Two volumes bound as one. MR 1255973
Vadim M. Adamjan and Heinz Langer, Spectral properties of a class of rational operator valued functions, J. Operator Theory 33 (1995), no. 2, 259–277. MR 1354980
Vadim Adamjan, Heinz Langer, and Christiane Tretter, Existence and uniqueness of contractive solutions of some Riccati equations, J. Funct. Anal. 179 (2001), no. 2, 448–473. MR 1809118, DOI 10.1006/jfan.2000.3680
Sergio Albeverio, Konstantin A. Makarov, and Alexander K. Motovilov, Graph subspaces and the spectral shift function, Canad. J. Math. 55 (2003), no. 3, 449–503. MR 1980611, DOI 10.4153/CJM-2003-020-7
Constantin Apostol, Ciprian Foias, and Norberto Salinas, On stable invariant subspaces, Integral Equations Operator Theory 8 (1985), no. 6, 721–750. MR 818331, DOI 10.1007/BF01213789
Rajendra Bhatia, Chandler Davis, and Paul Koosis, An extremal problem in Fourier analysis with applications to operator theory, J. Funct. Anal. 82 (1989), no. 1, 138–150. MR 976316, DOI 10.1016/0022-1236(89)90095-5
Rajendra Bhatia, Chandler Davis, and Alan McIntosh, Perturbation of spectral subspaces and solution of linear operator equations, Linear Algebra Appl. 52/53 (1983), 45–67. MR 709344, DOI 10.1016/0024-3795(83)80007-X
John Daughtry, Isolated solutions of quadratic matrix equations, Linear Algebra Appl. 21 (1978), no. 1, 89–94. MR 485926, DOI 10.1016/0024-3795(87)90202-3
Chandler Davis, The rotation of eigenvectors by a perturbation, J. Math. Anal. Appl. 6 (1963), 159–173. MR 149309, DOI 10.1016/0022-247X(63)90001-5
Chandler Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970), 1–46. MR 264450, DOI 10.1137/0707001
P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389. MR 251519, DOI 10.1090/S0002-9947-1969-0251519-5
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Vadim Kostrykin, Konstantin A. Makarov, and Alexander K. Motovilov, On a subspace perturbation problem, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3469–3476. MR 1991758, DOI 10.1090/S0002-9939-03-06917-X
Vadim Kostrykin, Konstantin A. Makarov, and Alexander K. Motovilov, Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach, Advances in differential equations and mathematical physics (Birmingham, AL, 2002) Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 181–198. MR 1991541, DOI 10.1090/conm/327/05814
Vadim Kostrykin, Konstantin A. Makarov, and Alexander K. Motovilov, A generalization of the $\tan 2\Theta$ theorem, Current trends in operator theory and its applications, Oper. Theory Adv. Appl., vol. 149, Birkhäuser, Basel, 2004, pp. 349–372. MR 2063758
H. Langer, A. Markus, V. Matsaev, and C. Tretter, A new concept for block operator matrices: the quadratic numerical range, Linear Algebra Appl. 330 (2001), no. 1-3, 89–112. MR 1826651, DOI 10.1016/S0024-3795(01)00230-0
R. McEachin, Closing the gap in a subspace perturbation bound, Linear Algebra Appl. 180 (1993), 7–15. MR 1206407, DOI 10.1016/0024-3795(93)90522-P