Criss-Cross Reduction of the Maslov Index and a Proof of the Yoshida-Nicolaescu Theorem

Abstract

We consider direct sum decompositions $\beta=\beta_{-}+\beta_{+}$ and $L=L_{-}+L_{+}$ of two symplectic Hilbert spaces by Lagrangian subspaces with dense embeddings $\beta_{-}\hookrightarrow L-$ and $L_{+}\hookrightarrow\beta_{+}$. We show that such criss-cross embeddings induce a continuous mapping between the Fredholm Lagrangian Grassmannians $\mathcal{F}\mathcal{L}_{\beta_{-}}(\beta)$ and $\mathcal{F}\mathcal{L}_{L_{-}}(L)$ which preserves the Maslov index for curves. This gives a slight generalization and a new proof of the Yoshida-Nicolaescu Spectral Flow Formula for families of Dirac operators over partitioned manifolds.