Journal of Mathematical Sciences

, Volume 194, Issue 6, pp 603–627 | Cite as

Operator Lipschitz functions and linear fractional transformations

  • A. B. Aleksandrov

It is known that the function \(t^2\sin\frac1t\) is an operator Lipschitz function on the real line \({\mathbb R}\). We prove that the function sin can be replaced by any operator Lipschitz function f with f(0) = 0. In other words, for every operator Lipschitz function f, the function \(t^2 f(\frac1t)\) is also operator Lipschitz if f(0) = 0. The function f can be defined on an arbitrary closed subset of the complex plane \({\mathbb C}\). Moreover, the linear fractional transformation \(\frac1t\) can be replaced by every linear fractional transformation ϕ. In this case, we assert that the function \(\dfrac{f\circ\varphi}{\varphi^{\,\prime}}\) is operator Lipschitz for every operator Lipschitz function f provided that f(ϕ( ∞ )) = 0. Bibliography: 12 titles.


Complex Plane Real Line Closed Subset Lipschitz Function Linear Fractional Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. B. Aleksandrov and V. V. Peller, “Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities,” Indiana Univ. Math. J., 59, 1451–1490 (2010).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. B. Aleksandrov and V. V. Peller, “Estimates of operator moduli of continuity,” J. Funct. Anal., 261, 2741–2796 (2011).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. B. Aleksandrov and V. V. Peller, “Operator and commutator moduli of continuity for normal operators,” Proc. London Math. Soc., 105, 82–85 (2012).MathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Yosida, Functional Analysis, Springer-Verlag, Berlin–Göttingen–Heidelberg (1965).MATHGoogle Scholar
  5. 5.
    B. E. Johnson and J. P. Williams, “The range of a normal derivation,” Pacific J. Math., 58, 105–122 (1975).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    H. Kamowitz, “On operators whose spectrum lies on a circle or a line,” Pacific J. Math., 20, 65–68 (1967).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    E. Kissin and V. S. Shulman, “On a problem of J. P. Williams,” Proc. Amer. Math. Soc., 130, 3605–3608 (2002).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    E. Kissin and V. S. Shulman, “Classes of operator-smooth functions. I. Operator-Lipschitz functions,” Proc. Edinb. Math. Soc. (2), 48, 151–173 (2005).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    E. Kissin and V. S. Shulman, “On fully operator Lipschitz functions,” J. Funct. Anal., 253, 711–728 (2007).MathSciNetMATHGoogle Scholar
  10. 10.
    E. Kissin, V. S. Shulman, and L. B. Turowska, “Extension of operator Lipschitz and commutator bounded functions,” Oper. Theory Adv. Appl., 171, 225–244 (2006).MathSciNetCrossRefGoogle Scholar
  11. 11.
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J. (1970).MATHGoogle Scholar
  12. 12.
    J. P. Williams, “Derivation ranges: open problems,” in: Topics Modern Operator Theory (Timisoara/Herculane, 1980), Operator Theory: Adv. Appl., 2, Birkhäuser, Basel–Boston, MA (1981), pp. 319–328.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

Personalised recommendations