  We extend to the infinite dimensional context the link between two completely
different topics recently highlighted by the authors: the classical eigenvalue
problem for real square matrices and the Brouwer degree for maps between
oriented finite dimensional real manifolds. Thanks to this extension, we solve
a conjecture regarding global continuation in nonlinear spectral theory that we
have formulated in a recent article. Our result (the ex conjecture) is applied
to prove a Rabinowitz type global continuation property of the solutions to a
perturbed motion equation containing an air resistance frictional force.
