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Mathematical Notes

, Volume 106, Issue 3–4, pp 481–487 | Cite as

On a Trace Formula for Functions of Noncommuting Operators

  • A. B. AleksandrovEmail author
  • V. V. PellerEmail author
  • D. S. PotapovEmail author
Article
  • 8 Downloads

Abstract

The main result of the paper is that the Lifshits-Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we show that, for pairs (A1, B1) and (A2, B2) of bounded self-adjoint operators with trace class differences A2-A1 and B2-B1, it is impossible to estimate the modulus of the trace of the difference f (A2, B2) - f (A1, B1) in terms of the norm of f in the Lipschitz class.

Keywords

trace trace class operators operators Lipschitz functions Lifshits-Krein trace formula 

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Notes

Funding

The research of the first author was supported in part by the Russian Foundation for Basic Research under grant 17-01-00607. The publication was prepared with the support of the “RUDN University Program 5-100.” The research of the third author was supported in part by ARC.

References

  1. 1.
    I. M. Lifshits, “On a problem of the theory of perturbations connected with quantum statistics,” Uspekhi Mat. Nauk 7 (1 (47)), 171–180 (1952).MathSciNetGoogle Scholar
  2. 2.
    M. G. Krein, “On the trace formula in perturbation theory,” Mat. Sb. 33 (75) (3), 597–626(1953).MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. G. Krein, “Perturbation determinants and a formula for traces of unitary and self-adjoint operators,” Dokl. Akad. Nauk SSSR 144 (2), 268–271 (1962) [Soviet Math. Dokl. 3, 707–719 (1962)].MathSciNetGoogle Scholar
  4. 4.
    M. G. Krein, “On some new investigations on the perturbation theory of self-adjoint operators,” in First Summer Mathematical School, I (Naukova Dumka, Kiev, 1964), pp. 103–187 [in Russian].Google Scholar
  5. 5.
    I. Ts. Gokhberg [Gohberg] and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators (Nauka, Moscow, 1965; AMS, Providence, RI, 1969).Google Scholar
  6. 6.
    M. Sh. Birman and M. Z. Solomyak [Solomjak], Spectral Theory of Selfadjoint Operators in Hilbert Space (Lan', SPb, 2010; transl. of the 1st ed. D. Reidel Publishing Co., Dordrecht, 1987).Google Scholar
  7. 7.
    Yu. B. Farforovskaya, “An example of a Lipschitzian function of selfadjoint operators that yields a nontrace-class increase under a trace-class perturbation,” in Investigations of Linear Operators and the Theory of Functions. III, in Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) (Izd. “Nauka”, Leningrad Otdel., Leningrad, 1972), Vol. 30, pp. 146–153.MathSciNetGoogle Scholar
  8. 8.
    V. V. Peller, “The Lifshits-Krein trace formula and operator Lipschitz functions,” Proc. Amer. Math. Soc. 144 (12), 5207–5215 (2016).MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. B. Aleksandrov and V. V. Peller, “Operator Lipschitz functions,” Uspekhi Mat. Nauk 71 (4(430)), 3–106 (2016) [Russian Math. Surveys, 71 (4), 605–702 (2016)].MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. V. Peller, “Hankel operators in the perturbation theory of unitary and self-adjoint operators,” Funktsional. Anal. Prilozhen. 19(2), 37–51 (1985) [Funct. Anal. Appl. 19(2), 111–123(1985)]MathSciNetzbMATHGoogle Scholar
  11. 11.
    V. V. Peller, “Hankel operators in the perturbation theory of unbounded self-adjoint operators,” in Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math. (Marcel Dekker, New York, 1990), Vol. 122, pp. 529–544.zbMATHGoogle Scholar
  12. 12.
    M. M. Malamud, H. Neidhardt, and V. V. Peller, “Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions,” Funktsional. Anal. Prilozhen. 51 (3), 33–55 (2017) [Funct. Anal. Appl. 51 (3), 185–203 (2017)].MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yu. B. Farforovskaya, “The connection of the Kantorovic-Rubinstein metric for spectral resolutions of selfadjoint operators with functions of operators,” Vestnik Leningrad. Univ. Ser. I Mat. Mekh. Astronom. 19, 94–97 (1968).MathSciNetGoogle Scholar
  14. 14.
    A. McIntosh, “Counterexample to a question on commutators,” Proc. Amer. Math. Soc. 29, 337–340 (1971).MathSciNetCrossRefGoogle Scholar
  15. 15.
    T. Kato, “Continuity of the map S → ⋎S⋎ for linear operators,” Proc. Japan Acad. 49, 157–160(1973).MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Sh. Birman and M. Z. Solomyak, “Remarks on spectral shift function,” in Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions. 6, in Zap. Nauchn. Sem. LOMI (Izd. “Nauka”, Leningrad sec, Leningrad, 1972), Vol. 27, pp. 33–46 [in Russian].MathSciNetzbMATHGoogle Scholar
  17. 17.
    M. M. Malamud, H. Neidhardt, and V. V. Peller, “Absolute continuity of spectral shift,” J. Funct. Anal. 276 (5), 1575–1621 (2019).MathSciNetCrossRefGoogle Scholar
  18. 18.
    B. Szökefalvi-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space (North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akademiai Kiado, Budapest 1970; Mir, Moscow, 1970).zbMATHGoogle Scholar
  19. 19.
    D. Potapov, F. Sukochev, and D. Zanin, “Krein's trace theorem revisited,” J. Spectr. Theory 4 (2), 415–430 (2014).MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. B. Aleksandrov, F. L. Nazarov, and V. V. Peller, “Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals,” Adv. Math. 295, 1–52 (2016).MathSciNetCrossRefGoogle Scholar
  21. 21.
    B. E. Johnson and J. P. Williams, “The range of a normal derivation,” Pacific J. Math. 58(1), 105–122(1975).MathSciNetCrossRefGoogle Scholar
  22. 22.
    E. Kissinand V. S. Shulman, “On a problem of J. P. Williams,” Proc. Amer. Math. Soc. 130 (12), 3605–3608 (2002).MathSciNetCrossRefGoogle Scholar
  23. 23.
    A. B. Aleksandrov, V. V. Peller, D. S. Potapov, and F. A. Sukochev, “Functions of normal operators under perturbations,” Adv. Math. 226 (6), 5216–5251 (2011).MathSciNetCrossRefGoogle Scholar
  24. 24.
    F. L. Nazarov and V. V. Peller, “Functions of perturbed n-tuples of commuting self-adjoint operators,” J. Funct. Anal. 266 (8), 5398–5428 (2014).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of SciencesSt. PetersburgRussia
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA
  3. 3.RUDN UniversityMoscowRussia
  4. 4.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia

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