Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier

TL;DR

This paper establishes an axiomatization of the Fitzpatrick-Pejsachowicz-Rabier degree for Fredholm operators, extending classical degree theories and providing a foundation for its uniqueness and applicability.

Contribution

It proves an axiomatization theorem for the degree of Fredholm operators of index zero, broadening the scope of classical degree theories.

Findings

Axiomatization of the Fitzpatrick-Pejsachowicz-Rabier degree

Extension of uniqueness theorems to a wider class of operators

Facilitation of the degree's axiomatization via algebraic multiplicity and parity

Abstract

In this paper we prove an analogue of the uniqueness theorems of F\"uhrer and Amann and Weiss to cover the degree of Fredholm operators of index zero of Fitzpatrick, Pejsachowicz and Rabier, whose range of applicability is substantially wider than for the most classical degrees of Brouwer and Leray-Schauder. A crucial step towards the axiomatization of the Fitzpatrick-Pejsachowicz-Rabier degree is provided by the generalized algebraic multiplicity of Esquinas and L\'{o}pez-G\'{o}mez, $\chi$, and the axiomatization theorem of Mora-Corral. The latest result facilitates the axiomatization of the parity of Fitzpatrick and Pejsachowicz, $\sigma(\cdot,[a,b])$, which provides the key step for establishing the uniqueness of the degree for Fredholm maps.