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p -Schatten commutators of projections
Annals of Functional Analysis ( IF 1.2 ) Pub Date : 2021-02-19 , DOI: 10.1007/s43034-021-00116-x
Esteban Andruchow , María Eugenia Di Iorio y Lucero

Let \({\mathcal {H}}={\mathcal {H}}_+\oplus {\mathcal {H}}_-\) be a fixed orthogonal decomposition of the complex separable Hilbert space \({\mathcal {H}}\) in two infinite-dimensional subspaces. We study the geometry of the set \({\mathcal {P}}^p\) of selfadjoint projections in the Banach algebra

$${\mathcal {A}}^p=\{A\in {\mathcal {B}}({\mathcal {H}}): [A,E_+]\in {\mathcal {B}}_p({\mathcal {H}})\},$$

where \(E_+\) is the projection onto \({\mathcal {H}}_+\) and \({\mathcal {B}}_p({\mathcal {H}})\) is the Schatten ideal of p-summable operators (\(1\le p <\infty\)). The norm in \({\mathcal {A}}^p\) is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space \({\mathcal {P}}^p\) is shown to be a differentiable \(C^\infty\) submanifold of \({\mathcal {A}}^p\), and a homogeneous space of the group of unitary operators in \({\mathcal {A}}^p\). The connected components of \({\mathcal {P}}^p\) are characterized, by means of a partition of \({\mathcal {P}}^p\) in nine classes, four discrete classes, and five essential classes: (1) the first two corresponding to finite rank or co-rank, with the connected components parametrized by these ranks; (2) the next two discrete classes carrying a Fredholm index, which parametrizes their components; (3) the remaining essential classes, which are connected.



中文翻译:

p-投影的夏滕换向器

\({\ mathcal {H}} = {\ mathcal {H}} _ + \ oplus {\ mathcal {H}} _- \\)是复数可分离Hilbert空间\({\ mathcal { H}} \)在两个无限维子空间中。我们研究Banach代数中自伴投影的集合\({\ mathcal {P}} ^ p \)的几何

$$ {\ mathcal {A}} ^ p = \ {A \ in {\ mathcal {B}}({\ mathcal {H}}):[A,E _ +] \ in {\ mathcal {B}} _ p ({\ mathcal {H}})\},$$

其中\(E _ + \)\({\ mathcal {H}} _ + \)的投影,而\({\ mathcal {B}} _ p({\ mathcal {H}})\)是Schatten理想的p -summable运营商(\(1 \文件p <\ infty \) )。在常态\({\ mathcal {A}} ^ P \)中的由上述分解给定运营商的矩阵条目的规范来定义。空间\ {{\ mathcal {P}} ^ p \)被显示为\ {{\ mathcal {A}} ^ p \)的可微分\(C ^ \ infty \)子流形和齐次空间\({\ mathcal {A}} ^ p \)中的of运算符组。\({\ mathcal {P}} ^ p \)的连接组件通过将\({\ mathcal {P}} ^ p \)划分为9个类,4个离散类和5个基本类来进行表征:(1)前两个对应于有限秩或同秩,这些等级已将连接的组件参数化;(2)带有弗雷德霍尔姆索引的接下来的两个离散类,将它们的组成参数化;(3)其余必不可少的类别,它们相互关联。

更新日期:2021-02-19
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