Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1511–1535 | Cite as

Lifshitz–Kreĭn Trace Formula for Hirsch Functional Calculus on Banach Spaces

  • A. R. MirotinEmail author


We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz–Kreĭn type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.


Spectral shift function Lifshitz–Kreĭn trace formula Hirsch functional calculus Negative operator Banach space Perturbation determinant 

Mathematics Subject Classification

Primary 47A56 Secondary 47A55 47B10 47L20 



The author would like to express his sincere gratitude to the referee for his very helpful comments and suggestions.


  1. 1.
    Aleksandrov, A.B., Peller, V.V.: Operator Lipschitz functions. Rus. Math. Surv. 71(4), 605–702 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aleksandrov, A.B., Peller, V.V.: Kreins trace formula for unitary operators and operator Lipschitz functions. Funct. Anal. Appl. 50(3), 167–175 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berg, C., Boyadzhiev, K., deLaubenfels, R.: Generation of generators of holomorphic semigroups. J. Austral. Math. Soc. (Ser. A) 55, 246–269 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Birman, M.S., Yafaev, D.R.: The spectral shift function. The papers of M. G. Kreĭn and their further development. Algebra i Analiz. 4, 1–44 (1992) (Russian). English transl.: St. Petersburg Math. J. 4, 833 – 870 (1993)Google Scholar
  5. 5.
    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  6. 6.
    Gesztesy, F., Makarov, K.A., Naboko, S.N.: The spectral shift operator. In: Dittrich, J., Exner, P., Tater, M. (eds.) Mathematical Results in Quantum Mechanics, Operator Theory: Advances and Applications, vol. 108, pp. 59–90. Birkhauser, Basel (1999)CrossRefGoogle Scholar
  7. 7.
    Gesztesy, F., Zinchenko, M.: Symmetrized perturbation determinants and applications to boundary data maps and Krein-type resolvent formulas. Preprint, arXiv:1007.4605v1 [math.SP]
  8. 8.
    Gokhberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. Nauka, Moscow (1965). Engliish transl. Amer. Math. Soc, Providence, R. I. (1969)Google Scholar
  9. 9.
    Hirsch, F.: Integrales de resolventes et calcul simbolique. Ann Inst. Fourier (Grenoble) 22(4), 239–264 (1972)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hirsch, F.: Transformation de Stieltjes et fonctions operant sur les potentiels abstraits. In: Faraut, J. (ed.) Theorie du Potentiel et Analyse Harmonique, Lecture Notes in Mathematics, vol. 404, pp. 149–163. Springer, Berlin (1974)CrossRefGoogle Scholar
  11. 11.
    Hirsch, F.: Familles d’operateurs potentiels. Ann Inst. Fourier (Grenoble) 25(3), 263–288 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hirsch, F.: Extension des proprietes des puissances fractionnaires. In: Hirsch, F., Makobodzki, G. (eds.) Seminaire de Theorie du Potentiel de Paris, Lecture Notes in Mathematics, vol. 563, no. 2, pp. 100–120. Springer, Berlin (1976)Google Scholar
  13. 13.
    Hirsch, F.: Domaines d’operateurs representes comme de integrales de resolventes. J. Func. Anal. 23, 199–217 (1976)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)CrossRefzbMATHGoogle Scholar
  15. 15.
    Komatsu, H.: Fractional powers of operators, III. J. Math. Soc. Jpn. 21, 205–220 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kreĭn, M.G.: On a trace formula in perturbation theory. Mat. Sbornik. 33, 597–626 (1953). (Russian)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Malamud, M., Neidhart, H.: Trace formulas for additive and non-additive perturbations. Adv. Math. 274, 736–832 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Malamud, M., Neidhart, H., Peller, V.: Analytic operator Lipschitz functions in the disc and a trace formula for functions of contractions. Funct. Anal. Appl. 51(3), 33–55 (2017). Preprint, arXiv:1705.07225 v1 [math. FA]MathSciNetCrossRefGoogle Scholar
  19. 19.
    Malamud, M., Neidhart, H., Peller, V.: A trace formula for functions of contractions and analytic operator Lipschitz functions. C. R. Acad. Sci. Paris Ser. I. 355, 806–811 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Martinez, C.C., Sanz, M.A.: The Theory of Fractional Powers of Operators. North-Holland Mathematics Studies, vol. 187. North-Holland Publishing Co., Amsterdam (2001)Google Scholar
  21. 21.
    Mirotin, A.R.: On the \(\cal{T}\)-calculus of generators for \(C_0\)-semigroups. Sib. Matem. Zh. 39(3), 571–582 (1998). English transl.: Sib. Math. J. 39 (3), 493–503 (1998)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mirotin, A.R.: Multidimensional \(\cal{T}\)-calculus for generators of \(C_0\) semigroups. Algebra i Analiz. 11(2), 142–170 (1999). English transl.: St. Petersburg Math. J. 11 (2), 315–335 (1999)Google Scholar
  23. 23.
    Mirotin, A.R.: The inverse of operator monotonic functions of negative operators on Banach spaces. Trudy Inst. Mat. (Minsk). 12(1), 104–108 (2004). (Russian)Google Scholar
  24. 24.
    Mirotin, A.R.: On multidimensional Bochner–Phillips functional calculus. Probl. Fiz. Mat. Tekh. 1(1), 63–66 (2009). (Russian)zbMATHGoogle Scholar
  25. 25.
    Mirotin, A.R.: Criteria for analyticity of subordinate semigroups. Semigroup Forum 78(2), 262–275 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mirotin, A.R.: On some properties of the multidimensional Bochner–Phillips functional calculus. Sib. Mat. Zhurnal. 52(6), 1300–1312 (2011). English transl.: Siberian Mathematical Journal. 52 (6), 1032–1041 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mirotin, A.R.: On joint spectra of families of unbounded operators. Izv. Ross. Akad. Nauk Ser. Mat. 79(6), 145–170 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mirotin, A.R.: On some functional calculus of closed operators on Banach space. III. Some topics in perturbation theory. Izvest. VUZ. Mat. 12, 24–34 (2017). (Russian); English transl.: Russian Math. (Iz. VUZ) 61 (12), 19–28 (2017)Google Scholar
  29. 29.
    Mirotin, A.R.: Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations. Oper. Matrices 11, 199–217 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mirotin, A.R.: Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations. II. Oper. Matrices 12, 445–463 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mirotin, A.R.: Corrections and complements to my paper ”on a class of operator monotone functions of several variables”. Math. Notes 101, 1061–1065 (2017). Original Russian Text: Matematicheskie Zametki, 101, 944 – 948 ( 2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Naboko, S.: Estimates in operator classes for a difference of functions, from the pick class, of accretive operators. Funct. Anal. Appl. 24(3), 187–195 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Peller, V.V.: The Lifshitz–Krein trace formula and operator Lipschitz functions. Proc. Am. Math. Soc. 144, 5207–5215 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Pustylnik, E.I.: On functions of a positive operator. Mat. Sb. 119(161), 32–47 (1982). English transl.: Math. USSR Sbornik. 47, 27–42 (1984)MathSciNetGoogle Scholar
  35. 35.
    Shilling, R., Song, R., Vondracek, Z.: Bernstein Functions. Theory and Applications. de Greyter, Berlin (2010)Google Scholar
  36. 36.
    Simon, B.: Trace Ideals and Their Applications, 2nd edn. American Mathematical Society, Providenc (2005)zbMATHGoogle Scholar
  37. 37.
    Yafaev, D.R.: Mathematical Scattering Theory. Translations of Mathematical Monographs, vol. 105. American Mathematical Society, Providence (1992)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Programming TechnologiesF. Skorina Gomel State UniversityGomelBelarus

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