An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree
Pierluigi Benevieri
Universidade de São Paulo, São Paulo, BrasilAlessandro Calamai
Università Politecnica delle Marche, Ancona, ItalyMaria Patrizia Pera
Università degli Studi di Firenze, Florence, Italy
![An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-zaa-volume-43-issue-1.png&w=3840&q=90)
Abstract
Let be a compact vector field of class on a real Hilbert space . Denote by the open unit ball of and by the unit sphere. Given a point , consider the self-map of defined by
If is finite dimensional, then is an orientable, connected, compact differentiable manifold. Therefore, the Brouwer degree, is well defined, no matter what orientation of is chosen, assuming it is the same for as domain and codomain of . This degree may be considered as a modern reformulation of the Kronecker index of the map . Let denote the Brouwer degree of on with target . It is known that one has the equality
Our purpose is an extension of this formula to the infinite dimensional context. Namely, we will prove that
where denotes the Leray–Schauder degree and is the degree earlier introduced by M. Furi and the first author, which extends, to the infinite dimensional case, the Brouwer degree and the Kronecker index. In other words, here, we extend to the Leray–Schauder degree the boundary dependence property which holds for the Brouwer degree in the finite dimensional context.
Cite this article
Pierluigi Benevieri, Alessandro Calamai, Maria Patrizia Pera, An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree. Z. Anal. Anwend. 43 (2024), no. 1/2, pp. 169–197
DOI 10.4171/ZAA/1750