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## Banach Center Publications

1994 | 30 | 1 | 147-159
Tytuł artykułu

### Functions of operators and their commutators in perturbation theory

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper shows some directions of perturbation theory for Lipschitz functions of selfadjoint and normal operators, without giving precise proofs. Some of the ideas discussed are explained informally or for the finite-dimensional case. Several unsolved problems are mentioned.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
147-159
Opis fizyczny
Daty
wydano
1994
Twórcy
autor
• Department of Mathematics, Electrotechnical Institute of Communication, Nab. r., Moĭki 61, 191065 St. Petersburg, Russia
Bibliografia
• [1] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Nauka, Moscow, 1966 (in Russian).
• [2] M. Sh. Birman, L. S. Koplienko and M. Z. Solomyak, Estimates of the spectrum of the difference of fractional powers of selfadjoint operators, Izv. Vyssh. Uchebn. Zaved. Mat. 1975 (3) (154), 3-10 (in Russian).
• [3] M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals, in: Probl. Mat. Fiz. 1, Leningrad Univ., 1966, 33-67 (in Russian).
• [4] M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals II, in: Probl. Mat. Fiz. 2, Leningrad Univ., 1967, 26-60 (in Russian).
• [5] M. Sh. Birman and M. Z. Solomyak, Remarks on spectral shift functions, Zap. Nauchn. Sem. LOMI 27 (1972), 33-41 (in Russian).
• [6] M. Sh. Birman and M. Z. Solomyak, Operator integration, perturbation and commutators, ibid. 170 (1989), 34-66 (in Russian).
• [7] J. Bourgain, On the similarity problem for polynomially bounded operators on Hilbert space, Israel J. Math. 54 (1986), 224-241.
• [8] M. D. Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988), 529-533.
• [9] Yu. L. Daletskiĭ and S. G. Kreĭn, Formulas for differentiation with respect to parameters of functions of hermitian operators, Dokl. Akad. Nauk SSSR 76 (1951), 13-16 (in Russian).
• [10] E. B. Davies, Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. 37 (1988), 148-157.
• [11] Yu. B. Farforovskaya, An example of a Lipschitz function of a selfadjoint operator giving a non-nuclear increment under a nuclear perturbation, Zap. Nauchn. Sem. LOMI 39 (1974), 194-195 (in Russian).
• [12] Yu. B. Farforovskaya, An estimate of the norm ∥f(A) - f(B)∥ for selfadjoint operators A and B, ibid. 56 (1976), 143-162 (in Russian).
• [13] Yu. B. Farforovskaya, An estimate of the norm ∥f(A₁,A₂) - f(B₁,B₂)∥ for pairs of commuting selfadjoint operators, ibid. 135 (1984), 175-177 (in Russian).
• [14] Yu. B. Farforovskaya, Commutators of functions of operators in perturbation theory, in: Probl. Mat. Anal. 12 (1992), 234-247 (in Russian).
• [15] L. V. Kantorovich and G. Sh. Rubinshteĭn, On a space of completely additive functions, Vestnik Leningrad. Gos. Univ. 13 (7) (1958), 52-59 (in Russian).
• [16] T. Kato, Continuity of the map S → |S| for linear operators, Proc. Japan Acad. 49 (1973), 157-160.
• [17] F. Kittaneh, On Lipschitz functions of normal operators, Proc. Amer. Math. Soc. 94 (1985), 416-418.
• [18] M. G. Kreĭn, On a trace formula in perturbation theory, Mat. Sb. 33 (1953), 597-626 (in Russian).
• [19] F. Kunert, The Kantorovich-Rubinshteĭn metric and convergence of selfadjoint operators, Vestnik Leningrad. Gos. Univ. 20 (13) (3) (1965), 37-49 (in Russian).
• [20] A. McIntosh, Counterexample to a question on commutators, Proc. Amer. Math. Soc. 29 (1971), 337-340.
• [21] B. Mirman, A source of counterexamples in operator theory and how to construct them, Linear Algebra Appl. 169 (1992), 49-59.
• [22] R. Moore, An asymptotic Fuglede theorem, Proc. Amer. Math. Soc. 50 (1975), 138-142.
• [23] V. V. Peller, Hankel operators and differentiability properties of functions of selfadjoint (unitary) operators, preprint LOMI, Leningrad, 1984.
• [24] V. V. Peller, Hankel operators in the theory of perturbations of unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen. 19 (2) (1985), 37-51 (in Russian).
• [25] V. V. Peller, For which f does $A - B ∈ S_p$ imply that $f(A) - f(B) ∈ S_p$?, in: Oper. Theory: Adv. Appl. 24, Birkhäuser, 1987, 289-294.
• [26] D. Voiculescu, Asymptotically commuting finite rank unitary operators without commuting approximation, Acta Sci. Math. (Szeged) 45 (1983), 429-431.
Typ dokumentu
Bibliografia
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