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 Studies on Nonlinear Lipschitz Operators(III)------the glb-Lipschitz Constant

In thispaper, the glb-Lipschitz constant l(T)of nonlinear Lipschitz operator T is introduced. Itis shown that theconstant l(T) qualitatively characterizes the biggest perturbationradiusat which the nonlinear invertible operator T maintains itsinvertibility in the semi-normoperator space composed of all Lipschitziancontinuous operators. The obtained results areused to establish a``Nonlinear Perturbation Lemma', to define nonlinear conditionnumberand to extend linear operator approximation theory. We apply also theobtainedresults to construct an inclusion region of spectrum ofnonlinear Lipschitz operator whichis a generalization of the famousGerschgorin theorem.