Operator \(\theta \)-Hölder functions

  • J. HuangEmail author
  • F. Sukochev
Survey Article


An important problem stemming from perturbation theory concerns description and understanding of operator \(\theta \)-Hölder functions. This article presents a survey of recent developments concerning operator \(\theta \)-Hölder functions with respect to symmetric quasi-norms.


Double operator integrals Operator inequalities Operator \(\theta \)-Hölder functions Noncommutative \(L_p\)-spaces, \(p>0\) Noncommutative quasi-Banach symmetric spaces 

Mathematics Subject Classification

47A55 46L51 47A60 47B10 


  1. 1.
    Nikol’skaya, L., Farforovskaya, Y.: Hölder functions are operator-Hölder. Algebra i Analiz 22, 198–213 (2010). (Russian). English translation: St. Petersburg Math. J. 22, 657–668 (2011)zbMATHGoogle Scholar
  2. 2.
    Aleksandrov, A.B., Peller, V.V.: Hankel and Toeplitz–Schur multipliers. Math. Ann. 324, 277–327 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aleksandrov, A.B., Peller, V.V.: Functions of operators under perturbations of class \({\mathbf{S}}_p\). J. Funct. Anal. 258, 3675–3724 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aleksandrov, A.B., Peller, V.V.: Operator Hölder–Zygmund functions. Adv. Math. 224, 910–966 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Aleksandrov, A.B., Peller, V.V.: Operator Lipschitz functions (Russian). Uspekhi Matematicheskikh Nauk 71(430), 3–106 (2016). (translation in Russian Math. Surveys 71(4), 605–702 (2016))MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aleksandrov, A.B., Peller, V.V., Potapov, D., Sukochev, F.: Functions of normal operators under perturbations. Adv. Math. 226, 5316–5251 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ando, T.: Comparison of norms \(\Vert |f(A)-f(B)|\Vert \) and \(\Vert f(|A-B|)\Vert \). Math. Z. 197, 403–409 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press Inc., London (1988)zbMATHGoogle Scholar
  9. 9.
    Bhatia, R.: Some inequalities for norm ideals. Commun. Math. Phys. 111, 33–39 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)zbMATHCrossRefGoogle Scholar
  11. 11.
    Birman, M.S., Solomjak, M.Z.: Estimates for the difference of fractional powers of selfadjoint operators under unbounded perturbations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989) (Issled. Lineǐn. Oper. Teorii Funktsiǐ. 18, 120–145, 185; translation in J. Soviet Math. 61(2), 2018–2035 (1992))Google Scholar
  12. 12.
    Birman, M., Solomjak, M.: Spectral Theory of Self-adjoint Operators in Hilbert Space, p. 301. D. Reidel Publishing Company, Dordrecht (1986)Google Scholar
  13. 13.
    Birman, M.S., Solomjak, M.Z.: Double operator integrals in a Hilbert space. Integr. Equ. Oper. Theory 47, 131–168 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Birman, M.S., Koplienko, L.S., Solomjak, M.Z.: Estimates of the spectrum of a difference of fractional powers of selfadjoint operators. Izv. Vysš. Učebn. Zaved. Matematika 3(154), 3–10 (1975)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Caspers, M., Montgomery-Smith, S., Potapov, D., Sukochev, F.: The best constants for operator Lipschitz functions on Schatten classes. J. Funct. Anal. 267, 3557–3579 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Caspers, M., Potapov, D., Sukochev, F., Zanin, D.: Weak type estimates for the absolute value mapping. J. Oper. Theory 73(2), 361–384 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Caspers, M., Potapov, D., Sukochev, F., Zanin, D.: Weak type commutator and Lipschitz estimates: resolution of the Nazarov–Peller conjecture. Am. J. Math. 141, 593–610 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Czerwinska, M., Kaminska, A.: Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals. Commun. Math. 57, 45–122 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Daletskii, YuL, Krein, S.G.: Integration and differentiation of functions of Hermitian operators and application to the theory of perturbations. Trudy Sem Funktsion. Anal. Voronezh. Gos. Univ. 1, 81–105 (1956). (in Russian)MathSciNetGoogle Scholar
  20. 20.
    Davies, E.B.: Lipschitz continuity of functions of operators in the Schatten classes. J. Lond. Math. Soc. 37, 148–157 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1995)zbMATHCrossRefGoogle Scholar
  22. 22.
    de Pagter, B., Sukochev, F.A.: Differentiation of operator functions in non-commutative \(L_{p}\)-spaces. J. Funct. Anal. 212, 28–75 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    de Pagter, B., Sukochev, E.: Commutator estimates and \(R{\mathbb{R}}\)-flows in non-commutative operator spaces. Proc. Edinb. Math. Soc. 50(02), 293–324 (2019)zbMATHCrossRefGoogle Scholar
  24. 24.
    de Pagter, B., Witvliet, H., Sukochev, F.: Double operator integral. J. Funct. Anal. 192, 52–111 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    de Pagter, B., Sukochev, F.A., Witvliet, H.: Double operator integrals. J. Funct. Anal. 192(1), 52–111 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Dixmier, J.: Les algebres d’operateurs dans l’Espace Hilbertien, 2nd edn. Gauthier-Vallars, Paris (1969)zbMATHGoogle Scholar
  27. 27.
    Dodds, P., de Pagter, B., Sukochev, F.: Theory of noncommutative integration (unpublished manuscript)Google Scholar
  28. 28.
    Dodds, P., Dodds, T., Sukochev, F., Zanin, D.: Arithmetic-geometric mean and related submajorization and norm inequalities for \(\tau \)-measurable operators (submitted manuscript)Google Scholar
  29. 29.
    Dodds, P., Dodds, T.: On a Submajorization Inequality of T. Ando, Operator Theory: Advances and Applications, vol. 75. Birkhäuser Verlag, Basel (1995)zbMATHGoogle Scholar
  30. 30.
    Dodds, P., de Pagter, B.: Normed Köthe spaces: A non-commutative viewpoint. Indag. Math. 25, 206–249 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Dodds, P., Sukochev, F.: Submajorisation inequalities for convex and concave functions of sums of measurable operators. Positivity 13, 107–124 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Dodds, P., Dodds, T., de Pagter, B.: Fully symmetric operator spaces. Integr. Equ. Oper. Theory 15, 942–972 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Dodds, P., Dodds, T., de Pagter, B.: Noncommutative Köthe duality. Trans. Am. Math. Soc. 339, 717–750 (1993)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Dodds, P., Dodds, T., Sukochev, F., Tikhonov, O.: A non-commutative Yosida–Hewitt theorem and convex sets of measurable operators closed locally in measure. Positivity 9, 457–484 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Dodds, P., Dodds, T., Sukochev, F.: On \(p\)-convexity and \(q\)-concavity in non-commutative symmetric spaces. Integr. Equ. Oper. Theory 78, 91–114 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expos. Math. 24, 1–37 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Fack, T., Kosaki, H.: Generalized \(s\)-numbers of \(\tau \)-measurable operators. Pac. J. Math. 123, 269–300 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Hardy, G.H.: Weierstrass’s non-differentiable function. Trans. Am. Math. Soc. 17, 301–325 (1916)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Hiai, F.: Matrix analysis: matrix monotone functions, matrix means, and majorization. Interdiscip. Inf. Sci. 16, 139–248 (2010)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hiai, F., Nakamura, Y.: Distance between unitary orbits in von Neumann algebras, with appendix: generalized powers-Størmer inequality by H. Kosaki. Pac. J. Math. 138, 259–294 (1989)zbMATHCrossRefGoogle Scholar
  41. 41.
    Huang, J., Sukochev, F., Zanin, D.: Operator \(\theta \)-Hölder functions with respect to \(||\cdot ||_p\), \(0< p\le \infty \) (submitted manuscript)Google Scholar
  42. 42.
    Huang, J., Levitina, G., Sukochev, F.: Completeness of symmetric \(\varDelta \)-normed spaces of \(\tau \)-measurable operators. Stud. Math. 237(3), 201–219 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Huang, J., Sukochev, F., Zanin, D.: Logarithmic submajorization and order-preserving isometries, submitted. J. Funct. Anal. 278(4), 108352 (2020)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Kalton, N.: Quasi-Banach Spaces. Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1099–1130. North-Holland, Amsterdam (2003)zbMATHCrossRefGoogle Scholar
  45. 45.
    Kalton, N., Sukochev, F.: Symmetric norms and spaces of operators. J. Reine Angew. Math. 621, 81–121 (2008)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Kalton, N., Peck, N., Rogers, J.: An F-space Sampler, London Math. Soc. Lecture Note Ser., vol. 89. Cambridge University Press, Cambridge (1985)Google Scholar
  47. 47.
    Kato, T.: Continuity of the map \(S\rightarrow |S|\) for linear operator. Proc. Jpn. Acad. 49, 157–160 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Kittaneh, F.: Inequalities for the Schatten p-norm IV. Commun. Math. Phys. 104, 581–585 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Krein, M.G.: On some new studies in the perturbation theory of self-adjoint operators. First Mathematics Summer School. Part I [in Russian], pp. 103–187. Naukova Dumka, Kiev (1964)Google Scholar
  50. 50.
    Krein, S., Petunin, Y., Semenov, E.: Interpolation of linear operators, Trans. Math. Mon., vol. 54. AMS, Providence (1982)Google Scholar
  51. 51.
    Leschke, H., Sobolev, A., Spitzer, W.: Trace formulas for Wiener–Hopf operators with applications to entropies of free fermionic equilibrium states. J. Funct. Anal. 273, 1049–1094 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Lord, S., Sukochev, F., Zanin, D.: Singular traces: theory and applications. In: Studies in Mathematics, vol. 46. Walter de Gruyter, Berlin (2013)Google Scholar
  53. 53.
    Löwner, K.: Uber monotone Matrixfunktionen (German). Math. Z. 38, 177–216 (1934)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Marshall, A., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and its Applications. Springer Series in Statistics, 2nd edn. Springer, New York (2011)zbMATHCrossRefGoogle Scholar
  55. 55.
    Nazarov, F., Peller, V.: Lipschitz functions of perturbed operators. C. R. Math. Acad. Sci. Paris 347, 857–862 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Okada, S., Ricker, W., Sánchez Pérez, E.: Optimal domain and integral extension of operators acting in function spaces. In: Operator Theory, Advances and Applications, vol. 180. Birkhäuser, Basel, Switzerland (2008)Google Scholar
  58. 58.
    Peller, V.: Hankel operator in the perturbation theory of unbounded self-adjoint operators. In: Analysis and Partial Differential Equations. Lecture Notes in Pure Appl. math., vol. 122, pp. 529–544. Dekker, New York (1990)Google Scholar
  59. 59.
    Peller, V.: Hankel operators in the theory of perturbations of unitary and self-adjoint operators. Funktsional. Anal. i Prilozhen. 19(2), 37–51 (1985). ((in Russian). English transl.: Funct. Anal. Appl. 19, 111–126 (1985))MathSciNetCrossRefGoogle Scholar
  60. 60.
    Pietsch, A.: Operator ideals. Deutscher Verlag der Wissenschaften, Berlin (1978)zbMATHGoogle Scholar
  61. 61.
    Pisier, G., Xu, Q.: Non-commutative \(L^p\)-spaces, in Handbook of the Geometry of Banach spaces, pp. 1459–1517. North-Holland, Amsterdam (2003)zbMATHCrossRefGoogle Scholar
  62. 62.
    Potapov, D., Sukochev, F.: Unbounded Fredholm modules and double operator integrals. J. Reine Angew. Math. 626, 159–185 (2009)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Potapov, D., Sukochev, F.: Double operator integrals and submajorization. Math. Model. Nat. Phenom. 5, 317–339 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Potapov, D., Sukochev, F.: Operator-Lipschitz functions in Schatten-von Neumann classes. Acta Math. 207, 375–389 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Powers, R., Størmer, E.: Free states of the canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Ricard, É.: Fractional powers on noncommutative \(L_p\) for \(p<1\). Adv. Math. 333, 194–211 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Segal, I.: A non-commutative extension of abstract integration. Ann. Math. 57, 401–457 (1953)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Skripka, A.: Operator integration in noncommutative analysis, Notices Amer. Math. Soc. (to appear)Google Scholar
  69. 69.
    Skripka, A., Tomskova, A.: Multilinear Operator Integrals: Theory and Applications, Lecture Notes in Mathematics. Springer, Berlin (2019)zbMATHCrossRefGoogle Scholar
  70. 70.
    Sobolev, A.: Functions of self-adjoint operators in ideals of compact operators. J. Lond. Math. Soc. 95(2), 157–176 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Sobolev, A.: Quasi-classical asymptotics for functions of Wiener–Hopf operators: smooth versus non-smooth symbols. Geom. Funct. Anal. 27, 676–725 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Sukochev, F.: Completeness of quasi-normed symmetric operator spaces. Indag. Math. 25, 376–388 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Triebel, H.: Theory of function spaces II. Birkhäuser, Basel (1992)zbMATHCrossRefGoogle Scholar
  74. 74.
    van Hemmen, J.L., Ando, T.: An inequality for trace ideals. Commun. Math. Phys. 76, 143–148 (1980)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2020

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia

Personalised recommendations