Cone characterizations of positive semidefinite operators on a Hilbert space (Q1060383)

Let \((H,<\cdot,\cdot >)\) be a real Hilbert space, T be a bounded linear operator on H with adjoint \(T^*\) and K be a closed convex cone in H. T is said to be (I) PSD on K if \(<Tx,x>\geq 0(\forall x\in K),\) (2) PSD plus on K if \(<Tx,x>>0(\forall x\in K,x\neq 0)\). Define \(K^ T:=\{y\in H:<y,(T+T^*)x>\leq 0\forall x\in K\}.\) The main result of the paper is Theorem 3.1. Let S be a subspace of H. Suppose that (i) T is PSD plus on S, (ii) T is PSD on \(S^ T\) and (iii) \((T+T^*)S\) is closed. Then T is PSD on H. This result is proved by solving a linear complementarity problem. The authors do not know if Theorem 3.1 holds when S is a closed convex cone.