关于非线性Lipschitz算子的Sderlind猜想

On Sderlind Conjectures on Nonlinear Lipschitz Operators

设ｆ是以Ｌ（ｆ）为最小上界Ｌｉｐｓｃｈｉｔｚ常数、以ρ（ｆ）为谱域半径、以ｒ（ｆ）为Ｇｅｒｓｃｈｇｏｒｉｍ域半径的有限维非线性Ｌｉｐｓｃｈｉｔｚ算子．本文证明了“存在等价范数‖·‖使Ｌ（ｆ）＝ｒ（ｆ）”的Ｓｄｅｒｌｉｎｄ猜想；给出反例否定了Ｓｄｅｒｌｉｎｄ的另一个猜想：“存在等价范数‖·‖使Ｌ（ｆ）ｒ（ｆ）”（注意ｒ（ｆ）与ｒ（ｆ）的区别），同时也否定了“ε＞０，存在等价范数‖·‖ε使Ｌε（ｆ）ρ（ｆ）＋ε”的猜想．作为以上所获结论的应用，本文将有关Ｄａｕｇａｖｅｔ方程的相应结果推广到了非线性算子情形．

Let f:D C m C m be a nonlinear Lipschitz operator with the 1.u.b. Lipschitz constant L(f) , the radius of Gerschgorim region r(f) and the radius of spectral region ρ(f) . In this paper it is justified the Sderlind's conjecture that there is an equivalent norm ‖·‖ * such that L *(f)=r *(f) , and disproved his another conjecture “there is an equivalent norm ‖·‖ * such that L *(f)r(f) ”. The counterexample presented disproved also the conjecture “For any ε >0, there is an equivalent norm ‖·‖ ε so that L ε(f)ρ(f)+ε ” proposed by Wang and Xu (this journal, 38:5 (1995), 628-631). As an applications of the confirmed Soderlind's conjecture,some fundamental identities on Daugavent equation related to a linear operator are generalized to the nonlinear case.