A topological approach to unitary spectral flow via continuous enumeration of eigenvalues

Abstract

It is a well-known result of T. Kato that given a continuous one-parameter family of square matrices of a fixed dimension, the eigenvalues of the family can be chosen continuously. In this paper, we give an infinite-dimensional analogue of this result, which arises in the context of unitary spectral flow. This intuitive topological approach to unitary spectral flow via continuous enumeration of eigenvalues appears to be missing from the existing literature, and it is the purpose of the present paper to fill in the gap. It is also shown in this paper that the notion of continuous enumeration naturally leads to a variant of the celebrated theorem of Dold-Thom in algebraic topology.