算子子理想的考察及子理想迹的介绍
A survey on subideals of operators and an introduction to subideal-traces
摘要：
算子理想中的B（H）是很好地理解和利用，但理想中的他们只是最近才开始研究与1983年开创性的工作方和Radjavi，并继续与最近的两篇文章的作者本次调查。本文对这三篇文章所体现的这一研究进行了综述。次理想是J的双边理想（特殊性也称为J-理想），其中J是B（H）的任意理想。在这个术语中，我们也称J为B（H）-理想。这调查这三篇文章中，我们制定了一个完整的刻画所有J-理想所产生的集的基数严格小于的基数的连续。所以一个中心主题是次理想的生成集对它们的代数结构的影响。这个特征特别包括可数生成的J-理想。它是通过第一次推广到任意的主要J-理想的1983年工作方Radjavi谁确定的主要K（H）-理想也是B（H）-理想。在我们的研究中，一个关键的性质被证明是J内的B（H）-理想I的J-软性，即IJ = I，这是Kaftal-Weiss最近引入的B（H）-理想的K（H）-软性概念的推广，并且更早地被Mityagin和Pietsch用于巴拿赫空间。子理想的研究与系数约束下的初等算子的研究密切相关。在这里，我们还引入并研究了子理想迹的概念，其中经典迹（酉不变的线性泛函）对于不是B（H）-理想的子理想不需要有意义。
英文摘要：
Operator ideals in B(H) are well understood and exploited but ideals inside them have only recently been studied starting with the 1983 seminal work of Fong and Radjavi and continuing with two recent articles by the authors of this survey. This article surveys this study embodied in these three articles. A subideal is a two-sided ideal of J (for specificity also called a J-ideal) for J an arbitrary ideal of B(H). In this terminology we alternatively call J a B(H)-ideal. This surveys these three articles in which we developed a complete characterization of all J-ideals generated by sets of cardinality strictly less than the cardinality of the continuum. So a central theme is the impact of generating sets for subideals on their algebraic structure. This characterization includes in particular finitely and countably generated J-ideals. It was obtained by first generalizing to arbitrary principal J-ideals the 1983 work of Fong-Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. A key property in our investigation turned out to be J-softness of a B(H)-ideal I inside J, that is, IJ = I, a generalization of a recent notion of K(H)-softness of B(H)-ideals introduced by Kaftal-Weiss and earlier exploited for Banach spaces by Mityagin and Pietsch. This study of subideals and the study of elementary operators with coefficient constraints are closely related. Here we also introduce and study a notion of subideal-traces where classical traces (unitarily invariant linear functionals) need not make sense for subideals that are not B(H)-ideals.
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