Spectral flows of dilations of Fredholm operators

TL;DR

This paper establishes a connection between spectral flow, operator index, and K-theory for dilations of Fredholm operators, extending to Z2 indices of symmetric operators, thus linking spectral flow with topological invariants.

Contribution

It introduces a spectral flow for dilations of Fredholm operators and relates it to the operator's index, extending the concept to Z2 indices of symmetric operators.

Findings

Spectral flow equals the index of the operator.

Spectral flow is interpreted via K-theory of a mapping cone.

Extension to Z2 indices of symmetric Fredholm operators.

Abstract

Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow which is shown to be equal to the index of the operator. This purely operator theoretic result is interpreted in terms of the $K$-theory of an associated mapping cone. It is then extended to connect $Z_2$ indices of odd symmetric Fredholm operators to a $Z_2$-valued spectral flow.