## Tokyo Journal of Mathematics

### 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.

#### Article information

Source
Tokyo J. Math., Volume 24, Number 1 (2001), 113-128.

Dates
First available in Project Euclid: 19 October 2009

https://projecteuclid.org/euclid.tjm/1255958316

Digital Object Identifier
doi:10.3836/tjm/1255958316

Mathematical Reviews number (MathSciNet)
MR1844422

Zentralblatt MATH identifier
1038.53072

#### Citation

BOOSS-BAVNBEK, Bernhelm; FURUTANI, Kenro; OTSUKI, Nobukazu. Criss-Cross Reduction of the Maslov Index and a Proof of the Yoshida-Nicolaescu Theorem. Tokyo J. Math. 24 (2001), no. 1, 113--128. doi:10.3836/tjm/1255958316. https://projecteuclid.org/euclid.tjm/1255958316