论文标题
标准张量和核操作员
Norm-attaining tensors and nuclear operators
论文作者
论文摘要
给定两个Banach Spaces $ x $和$ y $，我们介绍并研究了核操作员$ \ Mathcal {n}（x，y）$的规范性概念，并在投射张量的产品空间$ x \ wideHat {\ otimes}_πy$。我们展示了以前的规范缔造都有的正面和负面例子。我们还研究了以$ \ Mathcal {n}（x，y）$以及$ x \ wideHat {\ otimes}_πy$密集的元素的元素类别是否达到规范的问题。我们证明，对于这两个概念，标准构成元素的密度都适用于大量的Banach Spaces $ x $和$ y $，尤其是涵盖了所有古典的Banach空间。尽管如此，我们提出了Banach Spaces $ x $和$ y $失败的近似属性，以使$ x \ wideHat中的元素类{\ otimes}_πy$，它达到其投影规范并不密集。我们还讨论了我们工作的一些关系和应用于整个论文中规范性运营商的经典理论。
Given two Banach spaces $X$ and $Y$, we introduce and study a concept of norm-attainment in the space of nuclear operators $\mathcal{N}(X,Y)$ and in the projective tensor product space $X \widehat{\otimes}_πY$. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in $\mathcal{N}(X,Y)$ and in $X\widehat{\otimes}_πY$ is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces $X$ and $Y$ which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces $X$ and $Y$ failing the approximation property in such a way that the class of elements in $X\widehat{\otimes}_πY$ which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.
