Integro-differential operators on vector bundles
Integro-differential operators on vector bundles
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向量丛上的积分微分算子
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发表时间：
1965
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通讯作者：
R. Seeley
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文献类型：
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作者：
R. Seeley
介绍。本文考虑了紧流形上向量束截面上的一类相当一般的算子，包括“光滑”微分算子和奇异积分算子。这类算子具有微分算子的许多性质，特别是椭圆算子。有两大优势推动了这一发展。首先，它导致了椭圆方程的熟悉结果的透明证明，关于正则性，Fredholm替代和特征函数展开；对于一个比微分算子更大的类。这些证据并不新鲜；相反，在微分算子的情况下所使用的一些技术，在这里作为整微分算子类的一般性质出现。大型系统的第二个优点是拓扑结构，本文没有详细介绍。微分算子的特征多项式的同伦（在光滑函数类中）可以被“提升”到所考虑的积分-微分算子类中的算子本身的同伦，但在微分算子类中（通常）不能。这对处理Gelfand b[6]提出的一些问题有重要帮助；关于指数的一些问题现在已经由Atiyah和Singer b[1]回答了。要找到符号和主要结果，可以阅读§1 -§3（证明除外），§6，以及其余部分的定理和推论的定义和陈述。本文组织如下。§1描述了/{”上众所周知的函数空间，以及它们上的某些运算符。§2描述奇异积分算子及其符号。§3将这个集合扩展到包含R '上的微分算子，以及可逆椭圆算子的逆。定义了算子A的符号c(A)，讨论了算子A在复合条件下的行为。§4考虑a在坐标变化下的行为。§5给出了泛函分析的一些必要引理。§6建立了向量束的符号，以及§1中函数空间束的类似符号。§7定义了紧流形X上向量束E各节上的奇异积分算子及其符号。如果A是一个奇异积分算子来自于一个束E的各部分
Introduction. This article considers a fairly general class of operators on sections of a vector bundle over a compact manifold, including the "smooth" differential operators and singular integral operators. The members of this class share many of the properties of differential operators, particularly the elliptic ones. Two general advantages have motivated the development. First, it leads to transparent proofs of the familiar results for elliptic equations, on regularity, the Fredholm alternative, and eigenfunction expansions; and for a larger class than the differential operators. These proofs are not new; rather some of the techniques used in the case of differential operators appear here as general properties of the class of integro-differential operators considered. A second advantage of the larger system, not extensively exploited in this article, is topological. Homotopies (in the class of smooth functions) of the characteristic polynomial of a differential operator can be "lifted" to homotopies of the operator itself in the class of integro-differential operators considered, but not (generally) in the class of differential operators. This is an important help in treating some questions raised by Gelfand [6] ; some of the questions concerning the index have now been answered by Atiyah and Singer [1]. To find the notation and main results, one can read §1—§3 (except for proofs), §6, and the definitions and statements of theorems and corollaries from the remaining sections. The paper is organized as follows. §1 describes the well-known function spaces on /{"that are involved, as well as certain operators on them. §2 describes the singular integral operators and their symbols. §3 extends this collection to one that contains the differential operators on R", as well as the inverses of the invertible elliptic operators. The symbol c(A) of an operator A is defined, and the behavior of a under composition of operators is discussed. §4 considers the behavior of a under coordinate changes. §5 gives some necessary lemmas from functional analysis. §6 establishes the notation for vector bundles, and the analogs for bundles of the function spaces of §1. §7 defines the singular integral operators on sections of a vector bundle E over a compact manifold X, and their symbols. If A is a singular integral operator from sections of one bundle E