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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
Bulletin

Abstract

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.

Keywords

nonlinear Lipschitz operator Holub theorem Daugavet equation invertibility of operator 

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References

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    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
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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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