$B(H)$ -交换子：历史回顾II及紧算子交换子的研究进展
$B(H)$-Commutators: A Historical Survey II and recent advances on commutators of compact operators
摘要：
续集\cite{gW05}，我们再次解决单一的换向器问题\cite{PT71}的Pearcy和Topping：是每一个紧凑的运营商一个单一的换向器的紧算子？通过集中在一个35岁的测试问题，这提出了在1976年由最后命名的作者和其他人：是否有任何严格的积极的运营商是单一的紧算子？后者，我们解决了这里肯定与安德森的基本建设\cite{jA77}构造紧算子，其交换子是一个秩一投影的一个适度的修改。此外，我们在这里提供了一个丰富的类，这样的严格正的运营商，是紧算子的分解器，并提出了一个问题，其余的。我们还解释了这些方法是如何相关的阶梯矩阵形式，其等效块三对角形式，和换向器问题的研究。特别是，我们提出了原来的测试问题和解决方案，导致消极的解决方案的Pearcy-Topping问题是否每一个迹类迹零运营商是一个换向器（或线性组合的换向器）的希尔伯特-施密特运营商。我们展示了这是如何从阶梯形式的考虑以及Larry Brown关于迹连接到理想\cite{lB94}的结果发展而来的，理想本身就是\cite的核心[第7节]{DFWW}。遗漏在\cite{gW05}的这个重要的35岁的测试问题是无意的，我们在本文中纠正。这段经文是从[同上]开始的，但可以独立于[同上]阅读。本文还在一些经典算子李代数的框架下讨论了自换位子方程$[X^*,X]=A$。Fan和Fong（1980）在紧算子的全代数上解决了这个问题，我们在这里对紧算子的复辛李代数和复半单李代数解决了这个问题。
英文摘要：
A sequel to \cite{gW05}, we address again the single commutator problem \cite{PT71} of Pearcy and Topping: Is every compact operator a single commutator of compact operators? by focusing on a 35 year old test question for this posed in 1976 by the last named author and others: Are there any strictly positive operators that are single commutators of compact operators? The latter we settle here affirmatively with a modest modification of Anderson's fundamental construction \cite{jA77} constructing compact operators whose commutator is a rank one projection. Moreover we provide here a rich class of such strictly positive operators that are commutators of compact operators and pose a question for the rest. We explain also how these methods are related to the study of staircase matrix forms, their equivalent block tri-diagonal forms, and commutator problems. In particular, we present the original test question and solution that led to the negative solution of the Pearcy-Topping question on whether or not every trace class trace zero operator was a commutator (or linear combination of commutators) of Hilbert-Schmidt operators. And we show how this evolved from staircase form considerations along with a Larry Brown result on trace connections to ideals \cite{lB94} which itself is at the core of \cite[Section 7]{DFWW}. The omission in \cite{gW05} of this important 35 year old test question was inadvertent and we correct that in this paper. This sequel starts where [ibid] left off but can be read independently of [ibid]. The present paper also has a section on self-commutator equations $[X^*,X]=A$ within the framework of some classical operator Lie algebras. That problem was solved by Fan and Fong (1980) for the full algebra of compact operators, and we solve it here for the complex symplectic Lie algebra of compact operators and for complex semisimple Lie algebras.
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