关于算子与其自换位子的关系
On the relation between an operator and its self-commutator
摘要：
我们的主要结果是一个定理说，一个有界算子$A$在希尔伯特空间属于一定的集合与它的自交换子$[A^*,A]$，如果$A-zI$可以近似为可逆算子的所有复数$z$。在假定$A-zI$属于可逆元集的单位连通分支的闭包的条件下，该定理在一般的实秩零的$C^*$ -代数中仍然成立.这一结果推广了几乎可换矩阵的Brown-Douglas-Fillmore定理和林华新定理。此外，它允许我们改进前者，并将后者扩展到无限秩和其他范数（包括矩阵空间上的Schatten范数）的算子。证明是基于一个抽象定理，它指出，一个正规元素的$C^*$ -代数的实秩零满足上述条件有一个决议的单位元与任何开放覆盖的频谱。
英文摘要：
Our main result is a theorem saying that a bounded operator $A$ on a Hilbert space belongs to a certain set associated with its self-commutator $[A^*,A]$, provided that $A-zI$ can be approximated by invertible operators for all complex numbers $z$. The theorem remains valid in a general $C^*$-algebra of real rank zero under the assumption that $A-zI$ belong to the closure of the connected component of unity in the set of invertible elements. This result implies the Brown--Douglas--Fillmore theorem and Huaxin Lin's theorem on almost commuting matrices. Moreover, it allows us to refine the former and to extend the latter to operators of infinite rank and other norms (including the Schatten norms on the space of matrices). The proof is based on an abstract theorem, which states that a normal element of a $C^*$-algebra of real rank zero satisfying the above condition has a resolution of the identity associated with any open cover of its spectrum.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
