On the degree for oriented quasi-Fredholm maps: its uniqueness and its effective extension of the Leray--Schauder degree

In a previous paper, the first and third author developed a degree theory for oriented locally compact perturbations of C1 Fredholm maps of index zero between real Banach spaces. In the spirit of a celebrated Amann-Weiss paper, we prove that this degree is unique if it is assumed to satisfy three axioms: Normalization, Additivity and Homotopy invariance. Taking into account that any compact vector field has a canonical orientation, from our uniqueness result we shall deduce that the above degree provides an effective extension of the Leray-Schauder degree.