On the definition of eigenvalues for nonlinear operators (Q1570952)

The ``naive'' definition of eigenvalue for a nonlinear operator \(F\) in a Banach space \(X\) as a scalar \(\lambda\) with the property that the nullset \(N(\lambda I-F)= \{x\in X:\lambda x-F(x)= 0\}\) contains some nonzero element is not suitable for many purposes. In the present paper, the authors propose some other notions of eigenvalue, the most appropriate being the following: a scalar \(\lambda\) is called ``connected eigenvalue'' if \(N(\lambda I-F)\) contains an unbounded connected subset. With this definition, several well-known results carry over from the linear case; for example, the union of \(0\) and the set of connected eigenvalues of a compact nonlinear operator is always a compact subset of the complex plane.