椭圆的

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 12.1 椭圆算子与弗雷德霍姆算子

若矩阵 $\sigma (D,\xi )$ 对每个 $x\in M$ 和每个 $\xi \in {\mathbb{R}}^m-\{0\}$ 都可逆，则称算子 $D$ 是椭圆的。显然，这一椭圆性定义仅当 $k=k^{\prime }$ 时才有意义。应当注意，复合算子 $D=D_1\circ D_2$ 的符号是各符号的复合，即 $\sigma (D,\xi )=\sigma (D_1,\xi )\sigma (D_2,\xi )$。这表明椭圆算子的复合仍是椭圆算子。一般地，椭圆算子的幂与根也是椭圆算子。

英文原文

If the matrix $\sigma (D,\xi )$ is invertible for each $x\in M$ and each $\xi \in {\mathbb{R}}^m-\{0\}$ , the operator $D$ is said to be elliptic. Clearly this definition makes sense only when $k=k^{\prime }$ . It should be noted that the symbol for a composite operator $D=D_1\circ D_2$ is a composite of the symbols, namely $\sigma (D,\xi )=\sigma (D_1,\xi )\sigma (D_2,\xi )$ . This shows that composites of elliptic operators are also elliptic. In general, powers and roots of elliptic operators are elliptic.

出处：12.1 椭圆算子与弗雷德霍姆算子 · node a-d208d65687212a82a7fd12ae · 源节点 SHA-256 ce71a83cff730d7b…