非线性Lipschitz连续算子的定量性质(Ⅲ)──glb-Lipschitz数

本文引进非线性Ｌｉｐｓｃｈｉｔｚ算子Ｔ的ｇｌｂ－Ｌｉｐｓｃｈｉｔｚ数ｌ（Ｔ），并证明：ｌ（Ｔ）定量刻画非线性Ｌｉｐｓｃｈｉｔｚ连续算子全体所构成的赋半范算子空间中可逆算子Ｔ保持可逆的最大扰动半径，因而具有特别重要意义。所获结果被应用来建立“非线性扰动引理”、非线性算子条件数、推广线性算子逼近理论和建立与矩阵理论中Ｇｅｒｓｃｈｇｏｒｉｎ圆盘定理对应的非线性Ｌｉｐｓｃｈｉｔｚ连续算子谱集的包含域。

Qualitative Stndies on Nonlinear Lipschitz Operators ( )-the glb-Lipschitz Constant

in this paper, the glb-Lipschitz constant l(T) of nonlinear Lipschitz oper-ator T is introduced. It is shown that the constant l(T) qualitatively characterizesthe biggest perturbation radius at which the nonlinear invertible operator T maintainsits invertibility in the semi-norm operator space composed of all Lipschitzian contin-uous operators. The obtained results are used to establish a “Nonlinear PerturbationLemma”, to define nonlinear condition number and to extend linear operator approx-imation theory. We apply also the obtained results to construct an inclusion regionof spectrum of nonlinear Lipschitz operator which is a generalization of the famousGerschgorin theorem.