Simplicial cones and the existence of shape-preserving cyclic operators

Let X denote a real Banach space, X* its dual space and V an n-dimensional subspace of X. Given a weak*-closed cone S* X*, we say that f X has shape if f,φ 0 for all φ S*. Let S X denote the cone of elements having shape. Suppose the linear operator leaves S invariant (i.e., ). We seek extensions P of to X that leave S invariant; i.e. P:X→V such that and PS S. We say that such an extension is shape-preserving. It is shown in Chalmers and Prophet [Rocky Mountain J. Math. 28 (1998) (3) 813] that, under certain conditions on S*, admits a shape-preserving extension if and only if S*V is simplicial. In this paper we characterize those operators for which it is necessary and sufficient that S*V be simplicial in order to admit a shape-preserving extension. This characterization involves the eigenstructure of .