On linear maps leaving invariant the copositive/completely positive cones

Abstract

The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices \(\cal{S}^{n}\) that leave invariant the closed convex cones of copositive and completely positive matrices (COP_n and CP_n). A description of an invertible linear map on \(\cal{S}^{n}\) such that L(CP_n) ⊂ CP_n is obtained in terms of semipositive maps over the positive semidefinite cone \(\cal{S}_{+}^{n}\) and the cone of symmetric nonnegative matrices \(\cal{N}_{+}^{n}\) for n ⩽ 4, with specific calculations for n = 2. Preserver properties of the Lyapunov map X ↦ AX + XA^t, the generalized Lyapunov map X ↦ AXB + B^tXA^t, and the structure of the dual of the cone π(CP_n) (for n ⩽ 4) are brought out. We also highlight a different way to determine the structure of an invertible linear map on \(\cal{S}^{2}\) that leaves invariant the closed convex cone \(\cal{S}_{+}^{2}\).