Operators on Hilbert Space

摘要：概括地说，这个项目的目的是增加我们对作用在复杂的、可分的无限维希尔伯特空间上的线性算子的了解。提出者计划利用提出者、他的合著者和他的学生最近获得的结果和技术，继续在先前的NSF拨款下所做的工作。特别地，作者研究了各类算子的不变子空间问题，如Lomonosov服从算子、次n正态算子、谱半径为1的压缩算子和幂有界算子。这些类比其他一些提出者在解决不变子空间问题中起到了重要作用的类更为普遍。Hilbert空间上的算子可以被认为是复矩阵的自然推广，这类矩阵目前在解决现实世界中的许多问题如信号处理、人口研究、各种类型的优化过程等中起着重要的作用。但计算、量子物理、核燃料处理等中的某些问题的自然数学背景是在Hilbert空间上的更一般的算子设置中。有充分的证据表明，从纯粹的数学研究中获得的关于这类算子的知识在解决上述类型的现实世界问题中经常是有用的。这个项目的目的是增加关于运营商的基础知识。

Abstract : In broad terms, the purpose of this project is to increase our knowledge aboutlinear operators acting on a complex, separable, infinite dimensional Hilbert space.The proposer plans to continue work that was done under prior NSF grants, usingresults and techniques that were obtained in the recent past by the proposer, hiscoauthors, and his students, to accomplish this. In particular, the proposer willwork on the invariant subspace problem for various classes of operators suchas the Lomonosov-amenable operators, the sub-n-normal operators, the contractionswith spectral radius one, and the power bounded operators. These classes are moregeneral than some others in which the proposer has been instrumental in the solutionof the invariant subspace problem. Operators on Hilbert space may be thought of as the natural generalization offinite complex matrices, and such matrices play an important role presently inthe solution of many problems in the real world such as signal processing, populationstudies, various types of optimization procedures, etc. But the natural mathematicalsetting in which to cast certain problems in computing, quantum physics, nuclearfuel processing, etc. is in the more general setting of operators on Hilbert space.And there is ample evidence to show that knowledge about such operators, obtainedfrom a purely mathematical investigation, very frequently is useful in resolvingreal-world problems of the sort mentioned above. This project aims to increasesuch basic knowledge about operators.