Raffaele Chiappinelli. Source: Annals of Functional Analysis, Advance publication, 10 pages.Abstract:
We consider continuous gradient operators $F$ acting in a real Hilbert space $H$ , and we study their surjectivity under the basic assumption that the corresponding functional $\langle F(x),x\rangle $ —where $\langle \cdot \rangle $ is the scalar product in $H$ —is coercive. While this condition is sufficient in the case of a linear operator (where one in fact deals with a bounded self-adjoint operator), in the general case we supplement it with a compactness condition involving the number $\omega (F)$ introduced by Furi, Martelli, and Vignoli, whose positivity indeed guarantees that $F$ is proper on closed bounded sets of $H$ . We then use Ekeland’s variational principle to reach the desired conclusion. In the second part of this article, we apply the surjectivity result to give a perspective on the spectrum of these kinds of operators—ones not considered by Feng or the above authors—when they are further assumed to be sublinear and positively homogeneous.