Spectral Sections

TL;DR

This paper extends the concept of spectral sections to non-compact base spaces, explores their applications in cobordism theorems for Dirac operators, and examines the necessity of Riesz continuity for their existence.

Contribution

It generalizes spectral section results to arbitrary base spaces and investigates conditions for their existence, including Riesz continuity requirements.

Findings

Spectral sections are extendable to non-compact base spaces.

Cobordism theorems are established for families of Dirac operators.

Riesz continuity is shown to be necessary for spectral sections in certain cases.

Abstract

The paper is devoted to the notion of a spectral section introduced by Melrose and Piazza. In the first part of the paper we generalize results of Melrose and Piazza to arbitrary base spaces, not necessarily compact. The second part contains a number of applications, including cobordism theorems for families of Dirac type operators parametrized by a non-compact base space. In the third part of the paper we investigate whether Riesz continuity is necessary for existence of a spectral section or a generalized spectral section. In particular, we show that if a graph continuous family of regular self-adjoint operators with compact resolvents has a spectral section, then the family is Riesz continuous.