Operator Functional Equations in Analysis

  • Hermann König
  • Vitali Milman
Part of the Fields Institute Communications book series (FIC, volume 68)


Classical operations in analysis and geometry as derivatives, the Fourier transform, the Legendre transform, multiplicative maps or duality of convex bodies may be characterized, essentially, by very simple properties which may be often expressed as operator equations, like the Leibniz or the chain rule, bijective maps exchanging products with convolutions or bijective order reversing maps on convex functions or convex bodies. We survey and discuss recent results of this type in analysis. The operations we consider act on classical spaces like C k -spaces or Schwartz spaces \(\mathcal{S}({\mathbb{R}}^{n})\). Naturally, the results strongly depend on the type of the domain and the image space.

Key words

Operator equations Leibniz rule Chain rule Multiplicative maps Fourier transforms 



Hermann König was partially supported by the Fields Institute and Vitali Milman was partially supported by the Minkowski Center at the University of Tel Aviv, by the Fields Institute, by ISF grant 387/09 and BSF grant 2006079.


  1. 1.
    J. Aczél, Lectures on Functional Equations and their Applications (Academic, New York, 1966)MATHGoogle Scholar
  2. 2.
    S. Alesker, S. Artstein-Avidan, D. Faifman, V. Milman, A characterization of product preserving maps with applications to a characterization of the Fourier transform, Illinois J. Math. 54, 1115–1132 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    S. Artstein-Avidan, D. Faifman, V. Milman, On multiplicative maps of continuous and smooth functions, in GAFA Seminar 2006–2010, Springer Lecture Notes in Math. 2050, 35–59 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Artstein-Avidan, H. König, V. Milman, The chain rule as a functional equation. J. Funct. Anal. 259, 2999–3024 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. Banach, Sur l’équation fonctionelle \(f(x + y) = f(x) + f(y)\). Fund. Math. 1, 123–124 (1920)Google Scholar
  6. 6.
    H. Goldmann, P. Šemrl, Multiplicative derivations on C(X). Monatsh. Math. 121, 189–197 (1996)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    H. König, V. Milman, A functional equation characterizing the second derivative. J. Funct. Anal. 261, 876–896 (2011)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    H. König, V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule, with an appendix by D. Faifman. J. Funct. Anal. 261, 1325–1344 (2011)MATHCrossRefGoogle Scholar
  9. 9.
    H. König, V. Milman, An operator equation generalizing the Leibniz rule for the second derivative, in GAFA Seminar 2006–2010, Springer Lecture Notes in Math. 2050, 279–299 (2012)CrossRefGoogle Scholar
  10. 10.
    H. König, V. Milman, An operator equation characterizing the Laplacian. Algebra Anal. 24 (2013), to appearGoogle Scholar
  11. 11.
    H. König, V. Milman, A note on operator equations describing the integral, J. Math. Physics, Analysis, Geometry 9, 57–58 (2013)Google Scholar
  12. 12.
    H. König, V. Milman, Rigidity and stability of the Leibniz and the chain rule. Proc. Stekl. Inst. Math. 280 (2013, to appear)Google Scholar
  13. 13.
    A.N. Milgram, Multiplicative semigroups of continuous functions. Duke Math. J. 16, 377–383 (1940)MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Mrčun, P. Šemrl, Multplicative bijections between algebras of differentiable functions. Ann. Acad. Sci. Fenn. Math. 32, 471–480 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    W. Sierpinski, Sur l’equations fonctionelle \(f(x + y) = f(x) + f(y)\). Fund. Math. 1, 116–122 (1920)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Mathematisches SeminarUniversität KielKielGermany
  2. 2.Department of Mathematics, School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations