---
abstract: '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 selfadjoint 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.'
author:
- 'Marek Izydorek, Joanna Janczewska and Nils Waterstraat'
title: 'The Maslov Index and the Spectral Flow - revisited'
---

Introduction
============

Let $\langle\cdot,\cdot\rangle$ be the Euclidean scalar product on $\mathbb{R}^{2n}$ and $\omega_0(\cdot,\cdot)=\langle J\cdot,\cdot\rangle$ the standard symplectic form, where

$$\begin{aligned}
\label{J}
J=\begin{pmatrix}
0&-I_n\\
I_n&0
\end{pmatrix}\end{aligned}$$

and $I_n$ denotes the identity matrix. Let us recall that an $n$-dimensional subspace $L\subset\mathbb{R}^{2n}$ is called Lagrangian if the restriction of $\omega_0$ to $L\times L$ vanishes. The set $\Lambda(n)$ of all Lagrangian subspaces in $\mathbb{R}^{2n}$ is called the *Lagrangian Grassmannian*. It can be regarded as a submanifold of the Grassmannian $G_n(\mathbb{R}^{2n})$ and so it has a canonical topology. In 