论文标题
关于巴拉克空间之间的准规范运算符
On quasi norm attaining operators between Banach spaces
论文作者
论文摘要
我们引入了弱化的范围实现概念，用于在BANACH空间之间有限的线性操作员，我们称之为\ emph {quasi Norm达到操作员}。 Banach Spaces $ x $和$ Y $之间的操作员$ t \ colon x \ longrightArrow y $如果有$ x $的序列$（x_n）n norm on Norm on norm in $ x $，则$ x $（tx_n）$会收敛到y $ in $ u \ in y $，in y $ u \ in y $ c in $ \ \ \ \ \ \ \ \ \ \ | c。从通常的意义上获得规范的操作员（即，在单位球中有一个点的构图等于操作员的标准的操作员）和紧凑的操作员满足此定义。本文的主要结果是，可以通过准标准实现操作员（即使是更强的定义版本）来近似强大的radon-nikodým运算符，例如弱紧凑的操作员，这对于达到规范的操作员而言并不适合。这使我们能够根据域空间和范围空间的准标准实现运算符的密度来给出rad-nikodým属性的特征，从而扩展了Bourgain和Huff的先前结果。我们还对准规范运算符的密集性提出了积极和负面的结果，以实用准确的规范为准的有限维度和反射性来表征，讨论了要获得准标准的准入操作员实际上是在规范的情况下，研究与相邻操作员的规范达到的关系，并呈现某些稳定性。我们用一些开放的问题完成了论文。
We introduce a weakened notion of norm attainment for bounded linear operators between Banach spaces which we call \emph{quasi norm attaining operators}. An operator $T\colon X \longrightarrow Y$ between the Banach spaces $X$ and $Y$ is quasi norm attaining if there is a sequence $(x_n)$ of norm one elements in $X$ such that $(Tx_n)$ converges to some $u\in Y$ with $\|u\|=\|T\|$. Norm attaining operators in the usual sense (i.e.\ operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and compact operators satisfy this definition. The main result of the paper is that strong Radon-Nikodým operators such as weakly compact operators can be approximated by quasi norm attaining operators (even by a stronger version of the definition), which does not hold for norm attaining operators. This allows us to give characterizations of the Radon-Nikodým property in term of the denseness of quasi norm attaining operators for both domain spaces and range spaces, extending previous results by Bourgain and Huff. We also present positive and negative results on the denseness of quasi norm attaining operators, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator, and present some stability properties. We finish the paper with some open questions.
