论文标题
弗雷德霍尔姆（Fredholm
Fredholm conditions for operators invariant with respect to compact Lie group actions
论文作者
论文摘要
Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P \in ψ^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i \to M$, $i = 0,1$, and let $α$ be an irreducible representation of the group $ g $。然后$ p $诱导$π_α（p）：h^s（m; e_0）_α\ to h^{s-m}（m; e_1）_α$之间的$α$ - 异型组件之间。我们证明，如果$ p $为{\ em thressressy $α$ - elliptic}，地图$π_α（p）$是弗雷德霍尔姆（Fredholm），则是根据$ p $的主要符号和$ g $在vector bundles $ e_i $ e_i $上定义的条件。
Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P \in ψ^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i \to M$, $i = 0,1$, and let $α$ be an irreducible representation of the group $G$. Then $P$ induces a map $π_α(P) : H^s(M; E_0)_α\to H^{s-m}(M; E_1)_α$ between the $α$-isotypical components. We prove that the map $π_α(P)$ is Fredholm if, and only if, $P$ is {\em transversally $α$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$.
