Banach空间算子与算子类理论的研究

项目名称： Banach空间算子与算子类理论的研究

项目编号： No.11301155

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 原江涛

作者单位： 河南理工大学

项目金额： 22万元

中文摘要： Banach空间上的算子与算子类理论是泛函分析中两个具有密切联系的重要研究方向，近年来，它们发展较快并且在量子物理等领域显示出越来越多的应用。本项目将利用Fredholm与局部谱理论、矩阵论、算子方程与算子不等式、算子单调函数理论等研究Banach空间上的一般算子与算子类的性质及其应用。主要研究内容为：（1）以Fredholm型算子为基础研究Banach空间上一般算子的谱映射定理、Weyl型定理等谱性质；(2)Banach空间上一般算子与算子类的基本性质，比如上三角矩阵表示、不变子空间与近似点谱、孤立谱点的似极点与Riesz幂等、包含关系与遗传性等性质；（3）利用算子单调函数理论等研究算子理论在量子信息理论等方面的应用。预期结果可望将Stampfli与Putnam等的一些经典结果密切联系起来并扩展到一般情形。

中文关键词： Banach空间算子；Hilbert空间算子；Fredholm算子；非正规算子；谱

英文摘要： The theory of operators on Banach space and classes of opertors are two important areas in functional analysis. Recently, there are great developments in the two areas with more and more applications, such as quantum physics. This project is to study Banach space operators, classes of operators and their applications. By using the Fredholm and local spectral theory, operator equations, operator inequalities, matrix analysis, and so on, the following will be considered: (1) the spectral properties, such as spectral mapping theorems and Weyl type theorems of operators including Fredholm operators; (2) the elementary properties of Banach space operators and classes of operators, such as, the upper triangular matrix representation, invariant subspace and approximate point spectrum, isolated spectral points and Riesz idempotent, inclusion relations, heredity, and so on; (3) Some applications to related areas such as quantum information theory. Some classical results by Stampfli and Putnam may be extented and linked to each other.

英文关键词： Banach space operator；Hilbert space operator；Fredholm operator；nonnormal operator；spectrum