A new topology on the space of unbounded selfadjoint operators and the spectral flow

TL;DR

This paper introduces a new, weaker topology on unbounded selfadjoint operators, establishing their connection to K-theory and defining a generalized spectral flow for continuous paths of Fredholm operators.

Contribution

It defines a novel topology on unbounded selfadjoint operators and links the space of Fredholm operators to K-theory, extending spectral flow concepts.

Findings

The new topology is weaker than the gap topology.

Selfadjoint Fredholm operators represent K^1 and K^0 functors.

Spectral flow is generalized for continuous paths.

Abstract

We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor $K^1$ from the category of compact spaces to the category of abelian groups and prove a similar result for $K^0$. We define the spectral flow of a continuous path of selfadjoint Fredholm operators generalizing the approach of Booss-Bavnek--Lesch--Phillips.