  For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex
domains $\Omega$ in ${\mathbb R}^N$, a characterization of the principal
eigenvalue associated to a $k$-convex and negative principal eigenfunction will
be given as the supremum over values of a spectral parameter for which
admissible viscosity supersolutions obey a minimum principle. The admissibility
condition is phrased in terms of the natural closed convex cone $\Sigma_k$ in
the space of symmetric N by N matrices, which is an elliptic set in the sense
of Krylov [Trans. AMS, 1995] and which corresponds to using $k$-convex
functions as admissibility constraints in the formulation of viscosity
subsolutions and supersolutions. Moreover, the associated principal
eigenfunction is constructed by an iterative viscosity solution technique,
which exploits a compactness property which results from the establishment of a
global H\"{o}lder estimate for the unique $k$-convex solutions of the
approximating equations.
