Functions of operators and their commutators in perturbation theory
Farforovskaya, Yu.
Banach Center Publications, Tome 29 (1994), p. 147-159 / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 1994-01-01
EUDML-ID : urn:eudml:doc:262700
@article{bwmeta1.element.bwnjournal-article-bcpv30z1p147bwm,
     author = {Farforovskaya, Yu.},
     title = {Functions of operators and their commutators in perturbation theory},
     journal = {Banach Center Publications},
     volume = {29},
     year = {1994},
     pages = {147-159},
     zbl = {0804.47016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv30z1p147bwm}
}
Farforovskaya, Yu. Functions of operators and their commutators in perturbation theory. Banach Center Publications, Tome 29 (1994) pp. 147-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv30z1p147bwm/

[000] [1] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Nauka, Moscow, 1966 (in Russian). | Zbl 0098.30702

[001] [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). | Zbl 0313.47020

[002] [3] M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals, in: Probl. Mat. Fiz. 1, Leningrad Univ., 1966, 33-67 (in Russian). | Zbl 0161.34602

[003] [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). | Zbl 0182.46202

[004] [5] M. Sh. Birman and M. Z. Solomyak, Remarks on spectral shift functions, Zap. Nauchn. Sem. LOMI 27 (1972), 33-41 (in Russian).

[005] [6] M. Sh. Birman and M. Z. Solomyak, Operator integration, perturbation and commutators, ibid. 170 (1989), 34-66 (in Russian).

[006] [7] J. Bourgain, On the similarity problem for polynomially bounded operators on Hilbert space, Israel J. Math. 54 (1986), 224-241. | Zbl 0631.47008

[007] [8] M. D. Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988), 529-533. | Zbl 0649.15005

[008] [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). | Zbl 0042.34602

[009] [10] E. B. Davies, Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. 37 (1988), 148-157. | Zbl 0648.47011

[010] [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).

[011] [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).

[012] [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).

[013] [14] Yu. B. Farforovskaya, Commutators of functions of operators in perturbation theory, in: Probl. Mat. Anal. 12 (1992), 234-247 (in Russian). | Zbl 0895.47006

[014] [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).

[015] [16] T. Kato, Continuity of the map S → |S| for linear operators, Proc. Japan Acad. 49 (1973), 157-160. | Zbl 0301.47006

[016] [17] F. Kittaneh, On Lipschitz functions of normal operators, Proc. Amer. Math. Soc. 94 (1985), 416-418. | Zbl 0549.47006

[017] [18] M. G. Kreĭn, On a trace formula in perturbation theory, Mat. Sb. 33 (1953), 597-626 (in Russian).

[018] [19] F. Kunert, The Kantorovich-Rubinshteĭn metric and convergence of selfadjoint operators, Vestnik Leningrad. Gos. Univ. 20 (13) (3) (1965), 37-49 (in Russian).

[019] [20] A. McIntosh, Counterexample to a question on commutators, Proc. Amer. Math. Soc. 29 (1971), 337-340. | Zbl 0217.45503

[020] [21] B. Mirman, A source of counterexamples in operator theory and how to construct them, Linear Algebra Appl. 169 (1992), 49-59. | Zbl 0757.15016

[021] [22] R. Moore, An asymptotic Fuglede theorem, Proc. Amer. Math. Soc. 50 (1975), 138-142. | Zbl 0294.47023

[022] [23] V. V. Peller, Hankel operators and differentiability properties of functions of selfadjoint (unitary) operators, preprint LOMI, Leningrad, 1984.

[023] [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).

[024] [25] V. V. Peller, For which f does A-BSp imply that f(A)-f(B)Sp?, in: Oper. Theory: Adv. Appl. 24, Birkhäuser, 1987, 289-294.

[025] [26] D. Voiculescu, Asymptotically commuting finite rank unitary operators without commuting approximation, Acta Sci. Math. (Szeged) 45 (1983), 429-431. | Zbl 0538.47003