全连续映射的双投影逼近及其应用

Double Projection Approximation of Completely Continuous Mappings and Its Application

本文证明了Hilbert空间X上的连续映射G∶X→X为全连续的充要条件是G有双投影逼近性质,即任给X的有界子集D与ε>0,存在X的有限维子空间E,使得‖PG(Px)—G(x)‖≤ε,(vx∈D),其中P∶X→E为正交投影.还给出了这一结果到全连续梯度向量场的Morse指标上的应用.

Let X be a Hilbert space.We say that a mapping G:X→X has the double projection approximation property if for every bounded subset Dof X and every ε>0 there exists a finite-dimensional subspace E of Xsuch that‖PG(P_x)-G_(x)‖≤ε,A_x∈Dwhere P:X→E is the orthogonal projection.We prove that a continuousmapping G:X→X is completely continuous if and only if G has the doubleprojection approximation property.We give an application of this resultto the Morse index for the completely continuous gradient vector fields.