Mathematical Sciences: Problems in Modern Analysis

9401582 Enflo提出者计划在希尔伯特空间上进行两个涉及有界算子的项目。第一个项目是研究算子循环向量的轨道。特别是，这些轨道的不同正则性。第二个项目涉及使用提出者最近发展的技术研究不变子空间的某些方面。这里的重点是拟幂零算子。Hilbert空间上的算子表示矩阵的无穷维推广。众所周知，给定一个矩阵，总有一个向量，使得矩阵作用在该向量上的迭代范围不是整个向量空间。本课题研究的是无限维情形下的类似问题。经过适当的表述，这个问题被称为不变子空间问题。它一直是算子理论研究的热点。这个问题是试图将运算符分解为建立锁，这是分析运算符结构的核心。***

9401582 Enflo The proposers plan to work on two projects involving bounded operators on Hilbert space. The first project is to study orbits of cyclic vectors of operators. In particular, different regularity properties of these orbits. The second project involves a study of some aspects of invariant subspaces using recently developed techniques of the proposers. The focus here is on quasinilpotent operators. Operators on Hilbert space represent an infinite dimensional generalization of matrices. It is a well known fact that given a matrix there is always a vector such that the span of iterates of the matrix action on this vector is not the whole vector space. This project is focused on the analogous question in the infinite dimensional case. Appropriately formulated, this question is called the invariant subspace problem. It has been the focus of much research on operator theory. The question is an attempt to decompose operators into building `locks and is central to the analysis of the structure of operators. ***