Mathematical Sciences: Problems in Modern Analysis

9401582 Enflo提议者计划研究两个涉及希尔伯特空间上有界算子的项目。第一个项目是研究算子的循环向量的轨道。特别是这些轨道的不同规律性。第二个项目涉及使用最近提出的技术研究不变子空间的某些方面。这里的重点是拟幂算子。希尔伯特空间上的算子表示矩阵的无限维推广。这是一个众所周知的事实，给定一个矩阵，总有一个向量使得这个矩阵作用于这个向量的迭代张成的空间不是整个向量空间。本项目主要研究无限维情况下的类似问题。适当地表述，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焦点。这个问题试图将操作符分解为构建锁，并且是分析操作符结构的核心。＊＊＊

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. ***