算子代数中的紧元及其相关问题

离散元，是指算子代数中谱至多包含可数个点的元。由离散元组成的闭理想中的元，具有经典算子理论中紧算子的相似性质，从某种意义上可以看成是“紧元”。本项目拟利用拓扑根和多值解析函数理论，研究以下问题：1、弱化的Wojtynski问题；2、内导子离散值域问题；3、拟幂零元的离散扰动问题；4、推广的Weyl型定理；5、离散根的表示定理等。这些问题来源于著名数学家Shulman、Turovskii和Wojtynskii等人的工作，会奠定离散元类似与紧算子在算子理论中的地位和作用。

In this project, the scattered element means the element with at most countable spectrum. The elements in the ideal consisting of scattered elements have some similar properties with the compact operators in operator theory, so we can call them "compact elements" in operator algebras. By topological radical theory and analytic multifunction theory, We hope we can answer the classical problems on scattered elements in operator algebras, such as 1. weeked Wojtynski's problem, 2. scattered range of inner derivation problem, 3. scattered perturbation of quasinilpotent elements problem, 4. generalized Weyl's theorem, and 5. representation theory for scattered radicals etc. These problems posed by some famous mathematicians like Shulman, Turovskii and Wojtynski, and they will prove that the scattered elements will have the same positions and functions as the compact operators in operator theory.