Fréchet Algebra Techniques for Boundary Value Problems on Noncompact Manifolds: Fredholm Criteria and Functional Calculus via Spectral Invariance
Fréchet Algebra Techniques for Boundary Value Problems on Noncompact Manifolds: Fredholm Criteria and Functional Calculus via Spectral Invariance
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非紧流形边值问题的 Fréchet 代数技术：Fredholm 准则和基于谱不变性的泛函微积分
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发表时间：
1999
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通讯作者：
E. Schrohe
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作者：
E. Schrohe
针对（可能）非紧流形上的边值问题，提出了一种Boutet de Monvel型微积分。它基于一类加权符号和Sobolev空间。如果底层流形是紧的，就可以恢复标准演算。证明了以下内容：1阶零型Green算子的代数G是L(H)的谱不变fr<s:1>子代数，H是一个合适的Hilbert空间，即，2由于微积分中存在降阶算子，所以只关注阶零型的元素是没有限制的。3 .基于符号模低阶符号的可逆性，给出了边值问题Fredholm性质的充分必要判据；4 .对于若干复变量中G的元素，有一个全纯泛函演算。
A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven: 1 The algebra G of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of L(H), H a suitable Hilbert space, i. e., 2 Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus. 3 There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and 4 There is a holomorphic functional calculus for the elements of G in several complex variables.