论文标题
在一组规范的操作员和强大的伯克霍夫 - 詹姆斯正交性上
On a set of norm attaining operators and the strong Birkhoff-James orthogonality
论文作者
论文摘要
继续研究有关伯克霍夫詹姆斯正交性和运营商规范的最新结果的研究，我们引入了一种经营者的调整后的bhatia-šemrl特性，该财产比Bhatia-šemrl属性弱。具有调整后的Bhatia-šemrl属性的一组运算符包含在标准属性的集合中，就像Bhatia-šemrl属性一样。众所周知，如果操作员的域空间$ x $具有radon-nikodým属性（如有限维空间），但对于某些经典空间，诸如$ c_0 $，$ c_1 $，$ l_1 [0,1] $和$ c [0,1] $ c [0,1] $ c [0,1] $，众所周知，具有Bhatia-šemrl属性的运算符集是规范密集的。与Bhatia-šemrl属性相反，我们表明，当域空间为$ C_0 $或$ L_1 [0,1] $时，具有调整后Bhatia-šemrl属性的运算符组为标准密集。此外，我们表明，在$ c [0,1] $上具有调整后的Bhatia-šemrl属性的功能集并非规范密集，但是对于任何紧凑的Hausdorff $ k $，此类集合都弱 - $*$ - 浓度。
Continuing the study of recent results on the Birkhoff-James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia-Šemrl property for operators which is weaker than the Bhatia-Šemrl property. The set of operators with the adjusted Bhatia-Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia-Šemrl property. It is known that the set of operators with the Bhatia-Šemrl property is norm-dense if the domain space $X$ of the operators has the Radon-Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $c_0$, $L_1[0,1]$ and $C[0,1]$. In contrast with the Bhatia-Šemrl property, we show that the set of operators with the adjusted Bhatia-Šemrl property is norm-dense when the domain space is $c_0$ or $L_1[0,1]$. Moreover, we show that the set of functionals having the adjusted Bhatia-Šemrl property on $C[0,1]$ is not norm-dense but such a set is weak-$*$-dense in $C(K)^*$ for any compact Hausdorff $K$.
