Isometric properties of elementary operators

We consider the elementary operator L, acting on the Hilbert-Schmidt Class C₂(H), given by L(T)=ATB, with A and B bounded operators on H. We establish necessary and sufficient conditions on A and B for L to be a 2-isometry or a 3-isometry. We derive sufficient conditions for L to be an n-isometry. We also give several illustrative examples involving the weighted shift operator on l₂ and the multiplication operator on the Dirichlet space.