Decomposition of Spectral flow and Bott-type Iteration Formula

TL;DR

This paper proves the cogredient invariance of spectral flow for continuous paths of Fredholm operators and derives a decomposition formula, leading to a generalized Bott-type iteration formula for linear Hamiltonian systems.

Contribution

It introduces a new invariance property of spectral flow and provides a decomposition formula, extending Bott-type iteration formulas for Hamiltonian systems.

Findings

Spectral flow is cogredient invariant.

Decomposition formula for spectral flow under matrix-like invariance.

Generalized Bott-type iteration formula for Hamiltonian systems.

Abstract

Let $A(t)$ be a continuous path of Fredhom operators, we first prove that the spectral flow $sf(A(t))$ is cogredient invariant. Based on this property, we give a decomposition formula of spectral flow if the path is invariant under a matrix-like cogredient. As applications, we give the generalized Bott-type iteration formula for linear Hamiltonian systems.