Simplicial cones and the existence of shape-preserving cyclic operators
Cyclic matrices, Shape-preserving operators, Simplicial cones
Linear Algebra and Its Applications
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 fX has shape if 〈f,φ〉≥0 for all φS*. 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 = In 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. © 2003 Elsevier Inc. All rights reserved.
Original Publication Date
DOI of published version
Chalmers, B. L.; Prophet, M. P.; and Ribando, J. M., "Simplicial cones and the existence of shape-preserving cyclic operators" (2003). Faculty Publications. 3224.