Criss-cross reduction of the Maslov index and a proof of the Yoshida-Nicolaescu theorem.
This paper is a continuation of a previous one of the first two authors [Tokyo J. Math. 21, 1--34 (1998; Zbl 0932.37063)]. Let \(\beta\) and \(L\) be symplectic Hilbert spaces, \(\beta=\beta_{-}+\beta_{+}\) and \(L=L_{-}+L_{+}\) direct sum decompositions by transversal pairs of Lagrangian subspaces and assume that there exist continuous, injective mappings \(i_{-}:\beta_{-}\to L_{-}\) and \(i_{+}:L_{+}\to \beta_{+}\) with dense images which are compatible with the symplectic structures. For a fixed Lagrangian subspace \(\lambda_0\) of \(\beta\), let \({\mathcal F}{\mathcal L}_{\lambda_0}(\beta):=\{\mu\subset\beta\;| \;\mu\;\text{Lagrangian subspace and}\;(\mu,\lambda_0)\;\text{Fredholm pair}\}\). The authors prove that such criss-cross embeddings induce a continuous mapping between \(\mathcal{FL}_{\beta_{-}}(\beta)\) and \(\mathcal{FL}_{L_{-}}(L)\) which preserves the Maslov index for any continuous curve. This gives a slight generalization and a new proof of the Yoshida-Nicolaescu spectral flow formula for families of Dirac operators over closed connected smooth Riemannian partitioned manifolds [\textit{T. Yoshida}, Ann. Math. (2) 134, 277--323 (1991; Zbl 0748.57002), \textit{L. Nicolaescu}, Duke Math. J. 80, 485--533 (1995; Zbl 0849.58064)]. In the Appendix the authors correct a statement which was made in the above mentioned paper of the first two authors and they also explain why the main results of that paper remain valid.
scientific article; zbMATH DE number 1660516