Spectral sections: two proofs of a theorem of Melrose-Piazza

TL;DR

This paper provides two independent proofs of a fundamental theorem relating spectral sections of self-adjoint Fredholm operator families to the vanishing of their analytic index, clarifying key aspects and enabling generalizations.

Contribution

It introduces two new proofs of the Melrose-Piazza theorem, enhancing understanding and extending its applicability in index theory.

Findings

Spectral sections exist if and only if the analytic index vanishes.

The proofs clarify the definition of the analytic index and trivializations of Hilbert bundles.

The results enable generalizations of the original theorem.

Abstract

Spectral sections of families of self-adjoint Fredholm operators were introduced by Melrose and Piazza for the needs of index theory. The basic result about spectral sections is a theorem of Melrose and Piazza to the effect that a family admits a spectral section if and only if its analytic index vanishes. The present paper is devoted to two proofs of this theorem. These proofs allow to generalize this theorem and to clarify some subtle aspects related to the definition of the analytic index and trivializations of Hilbert bundles. It is based on ideas of author's paper arXiv:2111.15081, but is largely independent from it.