Operational calculus and semi-groups of operators
Operational calculus and semi-groups of operators
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运算微积分和半算子群
DOI：
10.1007/bfb0085481
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
H. Komatsu
中科院分区：
文献类型：
--
作者：
H. Komatsu
运算微积分是否属于数学一直存在争议。据我们所知，每本运筹学教科书都是从对这个问题的讨论开始的。大约十年前，日本数学会修订《百科全书数学词典》时，主编伊藤K·伊藤教授就第二版征求了外国学者的意见。一位法国数学家在回复中写道：“运算微积分没有被提及的价值；它是分布的一个糟糕的替代品，而且离有用还很远。”顺便说一句，吉田教授是该文章的作者。相反，他非常喜欢运筹学。 J. Mukusiriski 出版了关于运算微积分的新基础的书 [24] 后，他立即将其翻译成日语。他著名的教科书《泛函分析》[34] 中有一部分专门介绍了 Mikusiriski 的理论。他还写了四篇论文 [35,37,38,39] 和一本书的两个版本，一个是日语版 [36]，另一个是英语版 [40]。他对运算微积分的喜爱可能来自于他的信念：好的数学不仅必须是美丽的，而且必须是有用的。我们想这也来自于他的经历。在 1936 年发表的博士论文 [30] 中，他证明了巴拿赫代数中嵌入的局部紧群是李群。实际上Banach代数是他和Nagumo[25]当时提出的。他们通过幂级数展开定义了逆 (1-A)-I、指数 exp A 和对数 log (1+ A)，并证明了巴拿赫代数中元素 A 的任何一致连续半群都是 exp tA。 Nagumo [25]进一步通过求解器的复杂积分获得了非零谱上紧算子的Jordan分解。然而，他们没有注意到五年后出现的盖尔范德表述[61]。此后，Yosida教授应用Gelfand表示获得了正规算子交换族的联立谱分解[31]等结果。虽然他在Hille-Yosida定理的证明中再次使用了幂级数展开式[32]，但在了解Hille的工作后[7]，他对运算微积分的复杂方法产生了兴趣，并写了几篇论文，其中包括一篇关于算子分数幂的论文[33]。我们的系列论文 [8,9,10,11,12,13] 继承了它。在本次演讲中，我们给出了运算微积分、算子半群和算子值正弦函数的复杂方法的统一理论。拉普拉斯变换方法被认为仅适用于指数增长函数。然而，我们可以通过将拉普拉斯变换的定义扩展到超函数来克服这个困难。这样我们就有了运算微积分的新基础。考虑到
It has always been controversial whether or not Operational Calculus is a mathematics. As far as we know every textbook on Operational Calculus starts with a discussion on this issue. When the Mathematical Society of Japan revised its Encyclopedic Dictionary of Mathematics about ten years ago, the chief editor Professor K. Ito asked opinions of foreign scholars about the then second edition. In his reply a French mathematician wrote" Operational Calculus has no value of being mentioned; it is a bad succedaneum of distributions and is very far from being useful." Incidentally Professor Yosida was the author of that item. He liked Operational Calculus very much on the contrary. As soon as J. Mukusiriski published the book [24] on a new foundation of Operational Calculus, he arranged its translation into Japanese. A section of his famous textbook Functional Analysis [34] is devoted to Mikusiriski's theory. He also wrote four papers [35, 37, 38, 39] and two versions of a book, one in Japanese [36] and the other in English [40]. His fondness for Operational Calculus comes probably from his belief that a good mathematics must be not only beautiful but also useful. We imagine that it also comes from his experience. In his doctoral thesis [30] published in 1936, he proved that a locally compact group embedded in a Banach algebra is a Lie group. Actually Banach algebras were introduced by him and Nagumo [25] at that time. They defined inverses (1-A)-I, exponentials exp A and logarithms log (1+ A) by power series expansions and proved among others that any uniformly continuous semi-group is exp tA for an element A in the Banach algebra. Nagumo [25] went further to obtain the Jordan decomposition of a compact operator at a spectrum different from zero by complex integration of resolvents. They did not notice, however, the Gelfand representation [61 which appeared five years later. Since then Professor Yosida applied the Gelfand representation to obtain the simultaneous spectral decomposition of a commutative family of normal operators [31] and other results.Although he employed again power series expansions in his proof of the Hille-Yosida theorem [32], he was interested in the complex method of Operational Calculus after he knew Hille's work [7], and wrote several papers including one [33] on fractional powers of operators. Our series of papers [8, 9, 10, 11, 12, 13] succeeded it. In this talk we give a unified theory of the complex method of operational calculus, semi-groups of operators and operator valued sine functions. The method of Laplace transforms is believed to apply only to functions of exponential growth. However, we can overcome this difficulty by extending the definition of Laplace transforms to hyperfunctions. Thus we have a new foundation of operational calculus. Considering the