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 ∥, ∥I −T ∥ }. 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.
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References
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Peng, J., Xu, Z. Nonlinear version of Holub’s theorem and its application. Chin. Sci. Bull. 43, 89–91 (1998). https://doi.org/10.1007/BF02883912
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DOI: https://doi.org/10.1007/BF02883912