Chinese Science Bulletin

, Volume 43, Issue 2, pp 89–91 | Cite as

Nonlinear version of Holub’s theorem and its application

  • Jigen Peng
  • Zongben Xu


Holub proved that any bounded linear operator T or −T defined on Banach space L1(μ) satisfies Daugavet equation 1 + ∥T ∥ = Max {∥I + T ∥, ∥IT ∥ }. Holub’s theorem is generalized to the nonlinear case: any nonlinear Lipschitz operatorf defined on Banach space l1 satisfies 1 + L(f) = Max {L(I +f), L(I−f)}, where L(f) is the Lipschitz constant off. The generalized Holub theorem has important applications in characterizing the invertibility of nonlinear operator.


nonlinear Lipschitz operator Holub theorem Daugavet equation invertibility of operator 


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  1. 1.
    Holub, J. R., Daugavet’s equation and operators on L1(μ),Proc. Amer. Math. Soc., 1987, 100: 295.CrossRefGoogle Scholar
  2. 2.
    Schmidt, K. D., Daugavet’s equation and orthomorphisms,Proc. Amer. Math. Soc., 1990, 108(4): 905.CrossRefGoogle Scholar
  3. 3.
    Weis, L., An elementary approach to Daugavet equation, inLect. Notes in Pure Appl. Math. 175, New York and Basel: Marcel Dekker Inc., 1996, 494.Google Scholar
  4. 4.
    Holub, J. R., A note on best approximation and invertibility of operators on uniformly convex banach space,Internat. J. Math. and Math. Sci., 1991, 14(3): 611.CrossRefGoogle Scholar
  5. 5.
    Soderlind, G.. Bounds on nonlinear operators in finite-dimensional Banach space,Numer. Math., 1986, 50: 27.CrossRefGoogle Scholar

Copyright information

© Science in China Press 1998

Authors and Affiliations

  • Jigen Peng
    • 1
  • Zongben Xu
    • 1
  1. 1.Research Center for Applied Mathematics, Institute for Information and System ScienceXi’an Jiaotong UniversityXi’anChina

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