【摘要】 在 Banach空间 X中 ,证明了线性 1-集压缩场 A:X→ X为零指标 Fredholm算子的等价条件是 Ker A的维数有限和 Rang A为闭集 ,并在 A为 Fredholm算子的假设下 ,证明了不变直和分解定理 ,即 A的非正特征值的代数重数和是有限的 .据此 ,在紧致李群 G连续作用下的 Banach空间 X中 ,讨论了等变 1-集压缩场 f∈ C1(X,X )的正则零点轨道 N =G(x0 )的指标问题 ,证明了计算公式 ind(f ,N ) =(- 1) ρχ(N ) ,其中χ(N )是 N的欧拉示性数 ,ρ是 f′(x0 )的负特征值的代数重数和更多还原

【Abstract】 For a Banach space Xand a linear1 - set- contractive field A:X→ X ,if Ker Ais fi- nite dimensional and Rang Ais closed,it can be proved that Ais a zero- index Fredholm s op- erator and the algebraic multiplicity sum of the nonpositive eigenvalues of A is finite,i.e.,X exists an A - invariantdirectsum decomposion X =X- A X0A X*A ,where X- A = λ,n>0 Ker (λI + A) n,X0A=∪n>0 Ker Anand X+A =∩λ,n≥ 0 Rang(λI + A) n.According to above results,it can be discussed,in the Banach space X acted by a compact Lie group G ,how to calculate the index of the regular zero- orbit N =G( x0 ) off∈ C1 ( X,X) ,wherefis an equivariant1 - set- contractive field and Rangf′( x0 ) is closed.The main result is the index formula ind( f , N ) =( - 1 )ρ(f′(x0 ) )χ( N) ,whereχ( N ) is the Euler s characteristic number of N and ρ( f′( x0 ) ) is the algebraic multiplicity sum of negative eigenvalues off′( x0 ) .更多还原