A spectral theory for locally convex algebras
A spectral theory for locally convex algebras
复制标题
局部凸代数的谱理论
DOI：
10.1112/plms/s3-15.1.399
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发表时间：
1965
影响因子：
1.8
通讯作者：
G. Allan
中科院分区：
文献类型：
--
作者：
G. Allan
本文的目的是给出局部凸代数一般理论的元素(见(1.1))，特别是研究这种代数元素的谱概念。其基本思想来源于Banach空间E上闭算子T的谱理论。在这种情况下，T的谱是复数A的集合，对其XI-T没有有界逆。如果A是一个局部凸代数，第一个问题是为A的有界元找到一个合适的定义。(2.1)中给出的定义似乎是一个相当明显的选择，我认为，从它产生的理论来看，这是合理的。特别地，定理3.8表明，目前谱的定义依赖于有界元的定义，这是相当自然的。Waelbroeck(8)在更特殊的背景下讨论了一些类似的概念；然而，有一个拟完全和可交换性的假设。就像赋范代数的情况一样，有一些结果是有趣的，不依赖于任何这样的假设。这种一般情况的方法是通过代数的局部凸性，它允许系统地使用弱拓扑。Waelbroeck在文(7)中讨论的代数完备在某种意义上比局部凸代数更一般，但也因为完备性假设而更特殊。在本文中，我们不试图讨论一个以上变量函数的泛函演算问题，这是(7)中的主要问题。
The purpose of this paper is to present the elements of a general theory of locally convex algebras (see (1.1)), and in particular to examine a concept of spectrum for elements of such algebras. The basic idea derives from the spectral theory of a closed operator T on a Banach space E. In this case, the spectrum of T is the set of complex numbers A for which XI—T has no bounded inverse.(See eg (6).) If A isi a locally convex algebra, the first problem is to find a suitable definition for bounded element of A. The definition given in (2.1) seems a fairly obvious choice which is, I suggest, justified by the theory which stems from it. In particular, Theorem 3.8 suggests that the present definition of spectrum, which depends on that of bounded element, is a reasonably natural one.Some similar ideas, in a more special context, have been discussed by Waelbroeck (8); there, however, there is an assumption of quasicompleteness and of commutativity. As in the case of normed algebras, there are results which are interesting independently of any such assumptions. The approach to this general case is through the local convexity of the algebra, which permits a systematic use of the weak topology.(See (3.5) in this context.) The algkbres h bornes completes discussed by Waelbroeck in (7) are in a sense more general than locally convex algebras but are also more special because of the completeness assumption. We do not attempt, in the present paper, to discuss the problem of a functional calculus for functions of more than one variable, which is the main concern of (7).