Classification of certain commutator ideals and the tensor product closure property (Q1123377)

The author studies the commutator ideal equations \([I,I]=I^ 2\) and \([I,J]=IJ\) for two-sided ideals I,J\(\subset L(H)\). He gives new results and surveys old ones with giving simplified proofs. A sufficient condition for \([I,J]=IJ\) to hold is that I, J be closed under tensor products and that [I,J] contains a rank one orthogonal projection (the latter condition being necessary). 12 pairs of ideals are given where the conditions are satisfied, using L(H), compact operators, Schatten p-class operators. There is an ideal properly containing the Hilbert-Schmidt operators and properly contained in \(\cup_{p>2}S_ p(H)\) for which \([I,J]=I^ 2\); this answers a question of Salinas.