A Topological Approach to Unitary Spectral Flow via Continuous Enumeration of Eigenvalues

TL;DR

This paper extends the classical finite-dimensional eigenvalue continuity result to infinite-dimensional settings, introducing a topological approach to unitary spectral flow that fills a notable gap in the literature.

Contribution

It provides the first infinite-dimensional analogue of Kato's eigenvalue continuity theorem, offering a new topological perspective on spectral flow.

Findings

Establishes a continuous eigenvalue enumeration in infinite dimensions

Introduces a topological framework for unitary spectral flow

Fills a gap in the mathematical literature on spectral flow

Abstract

It is a well-known result of T.\,Kato that given a continuous path of square matrices of a fixed dimension, the eigenvalues of the path can be chosen continuously. In this paper, we give an infinite-dimensional analogue of this result, which naturally arises in the context of the so-called unitary spectral flow. This provides a new approach to spectral flow, which seems to be missing from the literature. It is the purpose of the present paper to fill in this gap.