On a formula for the spectral flow and its applications

2008/01/26 by Benevieri, Pierluigi, Piccione, Paolo
#Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0801.4102

Abstract

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index.

Related

• The Spectral Flow of a Restriction to a Subspace and the Maslov Indices in a Symplectic Reduction
• Ordinary differential equations in Banach spaces and the spectral flow
• Weak Symplectic Functional Analysis and General Spectral Flow Formula
• Spectral Flow, Maslov Index and Bifurcation of semi-Riemannian Geodesics
• Spectral flow and iteration of closed semi-Riemannian geodesics