# Functions of operators and their commutators in perturbation theory

Banach Center Publications (1994)

- Volume: 30, Issue: 1, page 147-159
- ISSN: 0137-6934

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topFarforovskaya, Yu.. "Functions of operators and their commutators in perturbation theory." Banach Center Publications 30.1 (1994): 147-159. <http://eudml.org/doc/262700>.

@article{Farforovskaya1994,

abstract = {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.},

author = {Farforovskaya, Yu.},

journal = {Banach Center Publications},

keywords = {perturbation theory for Lipschitz functions of selfadjoint and normal operators},

language = {eng},

number = {1},

pages = {147-159},

title = {Functions of operators and their commutators in perturbation theory},

url = {http://eudml.org/doc/262700},

volume = {30},

year = {1994},

}

TY - JOUR

AU - Farforovskaya, Yu.

TI - Functions of operators and their commutators in perturbation theory

JO - Banach Center Publications

PY - 1994

VL - 30

IS - 1

SP - 147

EP - 159

AB - 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.

LA - eng

KW - perturbation theory for Lipschitz functions of selfadjoint and normal operators

UR - http://eudml.org/doc/262700

ER -

## References

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- [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. Zbl0631.47008
- [8] M. D. Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988), 529-533. Zbl0649.15005
- [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). Zbl0042.34602
- [10] E. B. Davies, Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. 37 (1988), 148-157. Zbl0648.47011
- [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). Zbl0895.47006
- [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. Zbl0301.47006
- [17] F. Kittaneh, On Lipschitz functions of normal operators, Proc. Amer. Math. Soc. 94 (1985), 416-418. Zbl0549.47006
- [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. Zbl0217.45503
- [21] B. Mirman, A source of counterexamples in operator theory and how to construct them, Linear Algebra Appl. 169 (1992), 49-59. Zbl0757.15016
- [22] R. Moore, An asymptotic Fuglede theorem, Proc. Amer. Math. Soc. 50 (1975), 138-142. Zbl0294.47023
- [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 $A-B\in {S}_{p}$ imply that $f\left(A\right)-f\left(B\right)\in {S}_{p}$?, 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. Zbl0538.47003

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