An infinite dimensional version of the Kronecker index and its relation with the Leray--Schauder degree

Let $\f$ be a compact vector field of class $C^1$ on a real Hilbert space $\H$. Denote by $\B$ the open unit ball of $\H$ and by $\S = \partial\B$ the unit sphere. Given a point $q \notin \f(\S)$, consider the self-map of $\S$ defined by \[ \f_q^\partial(p) = \frac{\f(p)-q}{\|\f(p)-q\|}, \quad p \in \S. \] If $\H$ is finite dimensional, then $\S$ is an orientable, connected, compact differentiable manifold. Therefore, the Brouwer degree, $\deg_{Br}(\f_q^\partial)$ is well defined, no matter what orientation of $\S$ is chosen, assuming it is the same for $\S$ as domain and codomain of $\f_q^\partial$. This degree may be considered as a modern reformulation of the Kronecker index of the map $\f_q^\partial$. Let $\deg_{Br}(\f,\B,q)$ denote the Brouwer degree of $\f$ on $\B$ with target $q$. It is known that one has the equality \[ \deg_{Br}(\f,\B,q) = \deg_{Br}(\f_q^\partial). \] Our purpose is an extension of this formula to the infinite dimensional context. Namely, we will prove that \[ \deg_{LS}(\f,\B,q) = \deg_{bf}(\f_q^\partial), \] where $\deg_{LS}(\cdot)$ denotes the Leray--Schauder degree and $\deg_{bf}(\cdot)$ 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.