A note on operator equations describing the integral

H. König, V. Milman

Research output: Contribution to journalArticlepeer-review

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We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type fog + c=I (Tfog.Tg), f,g ∈ C1(ℝ), (1) where T: C1(ℝ) → C(ℝ) and where I is defined on C(ℝ). We consider suitable conditions on I and T such that (1) is well-defined and, after reformulating (1) as V(fog)=Tfog.Tg, f,g ∈ C1(ℝ) (2) with V: C1 (ℝ) → C(ℝ), give the general form of T, V and I. Simple initial conditions then guarantee that the derivative and the integral are the only solutions for T and I. We also consider an analogue of the Leibniz rule and study surjectivity properties there.

Original languageEnglish
Pages (from-to)51-58
Number of pages8
JournalJournal of Mathematical Physics, Analysis, Geometry
Issue number1
StatePublished - 2013


  • Chain rule
  • Integral
  • Leibniz rule
  • Operator equation


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