We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of self-adjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller’s theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.

中文翻译：

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重新审视了Maslov指数和光谱流

我们给出一个著名的Cappell，Lee和Miller定理的基本证明，该定理将一对Lagrangian子空间路径的Maslov索引与自伴一阶算子的相关路径的谱流相关联。我们特别注意后一个算子路径的连续性，在此我们考虑希尔伯特空间上所有封闭算子集的差距度量。最后，我们从Cappell，Lee和Miller定理中获得了线性哈密顿系统的谱流公式，该公式概括了Hu和Portaluri的最新结果。