AI 中文总结
该研究证明无穷维Fredholm算子群的若干经典滤过属于Fredholm $\Delta$-滤过，以$\GLK(\E)$的一般线性群典范包含链为例验证了维数序列为$n^2$，补充了经典Fredholm流形文献未涉及的非逐维递增滤过结论。
AI 中文摘要
在这篇短注中，我们证明了与某些摄动类相关的、无穷维Fredholm算子群的一些著名滤过，实际上是Fredholm $\Delta$-滤过（见下文定义5）。例如，设$\E$为可分无穷维实Hilbert空间。$\E$上所有为恒等算子的紧摄动的可逆算子构成的群$\GLK(\E)$，是建模在$\E$上的Hilbert Fredholm流形与丛的结构群\cite{ElwTr, Ksch, Mkhr}。利用一组标准正交基，存在一般线性群的典范包含关系：$$\GL(1) \subset \cdots \subset \GL(n) \subset \GL(n+1) \subset \cdots \subset \GL(\infty) = \varinjlim \GL(n) \subset \GLK(\E).$$我们证明这是Fredholm流形$\GLK(\E)$的一个Fredholm $\Delta$-滤过，其维数序列为$\Delta(n) = \dim(\GL(n)) = n^2$。由于滤过维数仅增加1的刚性约束，该结论在经典Fredholm流形文献中未被讨论过。
英文摘要
In this short note, we show that some well-known filtrations of infinite dimensional groups of Fredholm operators associated to certain perturbation classes are, in fact, Fredholm $Δ$-filtrations (see Definition 5 below.) For example, let $\E$ be a separable infinite dimensional real Hilbert space. The group $\GLK(\E)$ of all invertible operators on $\E$ which are compact perturbations of the identity is the structure group for Hilbert Fredholm manifolds and bundles modeled on $\E$ \cite{ElwTr, Ksch, Mkhr}. Using an orthonormal basis, there are canonical inclusions of general linear groups: $$\GL(1) \subset \cdots \subset \GL(n) \subset \GL(n+1) \subset \cdots \subset \GL(\infty) = \varinjlim \GL(n) \subset \GLK(\E).$$ We show this is a Fredholm $Δ$-filtration of the Fredholm manifold $\GLK(\E)$ with dimension sequence $Δ(n) = \dim(\GL(n)) = n^2$, which was not discussed in the classical Fredholm manifold literature because of the rigid constraint that the dimensions of a filtration increase only by one.
