what follows, we denote by $I$ the unit interval $[0,1]$. The Maslov index $\mu_{Mas}(\gamma_1,\gamma_2)$ assigns to any pair of paths $\gamma_1, \gamma_2:I\rightarrow\Lambda(n)$ an integer which, roughly speaking, is the total number of non-trivial intersections of the Lagrangian spaces $\gamma_1(\lambda)$ and $\gamma_2(\lambda)$ whilst the parameter $\lambda$ travels along the interval $I$. There are several different approaches to the Maslov index and here we just want to mention [@Maslov], [@Bott], [@Duistermaat], [@Sternberg], [@RobbinMaslov] and [@Zehnder], which is far from being exhaustive. Cappell, Lee and Miller introduced in [@Cappell] four different ways to define the Maslov index and showed that they are all equivalent. They first construct the Maslov index geometrically by using a stratification of $\Lambda(n)$ and intersection theory from differential topology following [@Sternberg]. Their approach also yields a uniqueness theorem for the Maslov index characterising this invariant uniquely by six axioms. The uniqueness theorem is then used to show that the Maslov index can alternatively be defined by determinant line bundles, $\eta$-invariants and the spectral flow, respectively.\
In this paper we focus on the latter invariant and aim to give a more elementary proof of the equality of the Maslov index and the spectral flow of a path of operators as 