The Maslov index in weak symplectic functional analysis

TL;DR

This paper rigorously develops the theory of the Maslov index for weak symplectic Banach spaces, extending its properties and highlighting new features in the context of functional analysis and geometry.

Contribution

It introduces a rigorous framework for the Maslov index in weak symplectic Banach spaces, including its basic properties and novel aspects.

Findings

Defined the Maslov index for paths of Fredholm pairs of Lagrangian subspaces

Established fundamental properties of the Maslov index in this setting

Highlighted new features arising in weak symplectic Banach spaces

Abstract

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.