When is a subspace of \(\mathcal B(\mathcal H)\) ideal? (Q1864534)

Let \(H\) be a real or complex Hilbert space. The main result states that a subspace \({\mathcal N}\) of \({\mathcal B}(H)\) is an ideal if \(P_1P_2A^*+ AP_1P_2\in{\mathcal N}\) for all \(A\in{\mathcal N}\) and all positive operators \(P_1,P_2\in{\mathcal B}(H)\).