论文标题
希尔伯特建筑群的痕迹
Traces for Hilbert Complexes
论文作者
论文摘要
我们研究了痕量操作员的新概念和痕量空间，用于抽象的希尔伯特建筑群。我们将痕量空间作为商空间/歼灭者介绍。我们表征相关跟踪操作员的内核和图像，并讨论痕量空间之间的二元关系。我们详细阐述的是，在有限的Lipschitz结构域上与欧几里得De Rham复合物相关的经典边界痕迹的许多特性植根于希尔伯特复合物的一般结构。我们获得了可以使用商空间/歼灭器配制的抽象痕迹希尔伯特建筑群。我们表明，如果希尔伯特建筑群（Hilbert Complex）与紧凑的举重操作员接纳稳定的“常规分解”，那么相关的痕量希尔伯特综合体是弗雷德霍尔姆（Fredholm）。在三维欧几里得空间中，将讨论抽象概念的化身和结果。
We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a Hilbert complex admits stable "regular decompositions" with compact lifting operators, then the associated trace Hilbert complex is Fredholm. Incarnations of abstract concepts and results in the concrete case of the de Rham complex in three-dimensional Euclidean space will be discussed throughout.
