论文标题
在Banach空间上的Fredholm操作员的扩展和Schur耦合后的等效性
Equivalence after extension and Schur coupling for Fredholm operators on Banach spaces
论文作者
论文摘要
扩展后的Schur耦合（SC）和等效性（EAE）是Banach空间上有限的操作员的重要关系。已有30年了，前者暗示了后者，但直到最近才构建一对eae而不是SC的Fredholm Operators，而不是SC。 在此结果的激励下，我们调查了Fred \ -HOLM运营商的EAE和SC何时重合。 EAE的Fredholm运营商具有相同的Fredholm指数。 Surprisingly, we find that for each integer $k$ and every pair of Banach spaces $(\mathcal{X},\mathcal{Y})$, either no pair of Fredholm operators of index~$k$ acting on $\mathcal{X}$ and $\mathcal{Y}$, respectively, is SC, or every pair of this kind which is EAE is also SC.因此，索引〜$ k $的Fredholm操作员的EAE和SC是否仅取决于基础Banach空间的几何形状$ \ Mathcal {X} $和$ \ Mathcal {Y} $，而不在操作员本身的属性。 我们通过引入两个捕获EAE和SC巧合的数值索引来量化这一发现，并提供了许多示例，以说明这些索引的可能值。值得注意的是，这包括一个示例，表明基于一对本质上无与伦比的Banach空间的Ter Horst等人的上述结果并未扩展到投影无与伦比的Banach空间。
Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC. Motivated by this result, we investigate when EAE and SC coincide for Fred\-holm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer $k$ and every pair of Banach spaces $(\mathcal{X},\mathcal{Y})$, either no pair of Fredholm operators of index~$k$ acting on $\mathcal{X}$ and $\mathcal{Y}$, respectively, is SC, or every pair of this kind which is EAE is also SC. Consequently, the question whether EAE and SC coincide for Fredholm operators of index~$k$ depends only on the geometry of the underlying Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, not on the properties of the operators themselves. We quantify this finding by introducing two numerical indices which capture the coincidence of EAE and SC, and provide a number of examples illustrating the possible values of these indices. Notably, this includes an example showing that the above-mentioned result of Ter Horst et al, which is based on a pair of essentially incomparable Banach spaces, does not extend to projectively incomparable Banach spaces.
