论文标题
在Banach空间之间达到规范运算符的一组剩余性
Residuality in the set of norm attaining operators between Banach spaces
论文作者
论文摘要
我们研究了Banach空间上达到规范功能的剩余性与剩余性以及在Banach空间之间达到规范运算符的密度之间的关系。我们的第一个主要结果是，如果$ c $是Banach Space $ x $的一个有界子集，它承认，它可以重新恢复满足，对于每个Banach Space $ y $，从$ x $ y $ $ x $到$ y $的每个$ t $ $ t $ th $ \ \ \ | tx \ | tx $ x $的$ x $ congens的$ c ge_的us y Is y Is y Is y Is y Is y Is y Is y Is y Is y Is y Is y Is y Is y Inje y y y $ c $ t $ c $ y $ c。 $ x^*$。这扩展了J. \ Bourgain和K.-S. \ Lau的先前结果。 $ c $是$ x $的单位球的特殊情况，在这种情况下，我们认为$ x^*$的规范在密集的子集中是可区分的，它改善了J. \ lindenstrauss的结果，我们甚至提供了一个例子，表明Lindenstrauss的结果并非最佳。在反向方向上，我们通过要求$g_δ$集的密度完全强烈地将运营商从$ x $暴露于$ y $，这是要求在$ x $上强烈暴露在$ y $或$ y $或$ y^*$上的条件，涉及rnp和涉及RNP的条件，并在涉及rnp的$ y $ y $ y $ y $ y $ y或$ y y y或y y $ y y^$ y^$ y^$^$上。这些结果包括示例，即即使达到规范的运算符的密度也未知。我们还表明，获得规范的运算符的剩余性意味着，只要域空间和范围空间的双重双重，绝对强烈地揭示了操作员的密度，从而扩大了功能的最新结果。最后，我们的结果找到了重要的应用，其中我们指出，我们解决了一个提出的开放问题，表明Lipschitz函数的独特效果来自Euclidean单位圆圈，无法使Lindenstrauss属性A。
We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if $C$ is a bounded subset of a Banach space $X$ which admit an LUR renorming satisfying that, for every Banach space $Y$, the operators $T$ from $X$ to $Y$ for which the supremum of $\|Tx\|$ with $x\in C$ is attained are dense, then the $G_δ$ set of those functionals which strongly exposes $C$ is dense in $X^*$. This extends previous results by J.\ Bourgain and K.-S.\ Lau. The particular case in which $C$ is the unit ball of $X$, in which we get that the norm of $X^*$ is Fréchet differentiable at a dense subset, improves a result by J.\ Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the $G_δ$ set of absolutely strongly exposing operators from $X$ to $Y$ by requiring that the set of strongly exposing functionals on $X$ is dense and conditions on $Y$ or $Y^*$ involving RNP and discreteness on the set of strongly exposed points of $Y$ or $Y^*$. These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications, among which we point out that we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A.
