  We consider contractions of complexified real cones, as recently introduced
by Rugh in [Rugh10]. Dubois [Dub09] gave optimal conditions to determine if a
matrix contracts a canonical complex cone. First we generalize his results to
the case of complex operators on a Banach space and give precise conditions for
the contraction and an improved estimate of the size of the associated spectral
gap. We then prove a variational formula for the leading eigenvalue similar to
the Collatz-Wielandt formula for a real cone contraction. Morally, both cases
boil down to the study of suitable collections of 2 by 2 matrices and their
contraction properties on the Riemann sphere.
