Simplicial cones and the existence of shape-preserving cyclic operators

Abstract

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 P:V→V leaves S invariant (i.e., P(S∩V)⊂S ). We seek extensions P of P to X that leave S invariant; i.e. P: X→ V such that P | V = P 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 *, P=I n admits a shape-preserving extension if and only if S * | V is simplicial. In this paper we characterize those operators P 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 P .