Fredholm operators associated with strongly pseudoconvex domains in Cn
Fredholm operators associated with strongly pseudoconvex domains in Cn
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Fredholm 算子与 Cn 中的强伪凸域相关
DOI：
10.1016/0022-1236(72)90007-9
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发表时间：
1972
影响因子：
1.7
通讯作者：
U. Venugopalkrishna
中科院分区：
文献类型：
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作者：
U. Venugopalkrishna
本文推广了Gohberg和Krien关于单位圆上Weiner-Hopf算子的指数定理。令 Ω 为 C n 中的强赝凸域，并假设 L 2 N (Ω) 为平方可积函数 f: Ω→ C N 的空间。令 H 2 N (Ω) 为 Ω 中全纯的所有 f ϵ L 2 N (Ω) 的子空间，并令 P: L 2 N (Ω)→ H 2 N (Ω) 为正交投影。令 s 为 Ω 上的连续 N× N 矩阵值函数，其在 Ω 中平滑，使得对于 z ϵ∂ Ω，det s (z)≠ 0，并令 S: H 2 N (Ω)→ L 2 N (Ω) 为由 S 1 f= sf 对于 f ϵ H 2 N (Ω) 定义的算子。进而证明算子S= PS 1 是Fredholm。获得指数通用公式的问题是开放的。然而，如果Ω是C n 中的开单位球，则证明算子S的索引等于(− 1) n次s对∂ Ω的限制，如果N⩾ n。这些结果与 Atiyah-Singer 指数定理有明显的联系。
This paper generalizes the index theorem of Gohberg and Krien on Weiner-Hopf operators on the unit circle. Let Ω be a strongly pseudoconvex domain in C n and suppose L 2 N (Ω) is the space of square integrable functions ƒ: Ω→ C N. Let H 2 N (Ω) be the subspace of all ƒ ϵ L 2 N (Ω) which are holomorphic in Ω and let P: L 2 N (Ω)→ H 2 N (Ω) be the orthogonal projection. Let s be a continuous, N× N matrix valued function on Ω which is smooth in Ω such that det s (z)≠ 0 for z ϵ∂ Ω, and let S: H 2 N (Ω)→ L 2 N (Ω) be the operator defined by S 1 ƒ= sƒ for ƒ ϵ H 2 N (Ω). It is then proved that the operator S= PS 1 is Fredholm. The problem of obtaining a general formula for index is open. However, if Ω is the open unit ball in C n, then it is proved that the index of the operator S is equal to (− 1) n time degree of the restriction of s to∂ Ω, if N⩾ n. These results have obvious points of contact with the Atiyah-Singer index theorem.