Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier

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.