The closure of the range of an elementary operator Original Research Article

Let B(H) denote the algebra of operators on a Hilbert space H, and let phi set membership, variant B(B(H)) be the elementary operator defined by phi(X) = AXB+CXD. A necessary condition for phi−1(0) circled plus phi(B(H)) = B(H) is that 0 is an isolated point of the spectrum σ(phi) of phi. We prove a sufficient condition for phi−1(0) circled plus phi(B(H)) = B(H). Applied to the case in which the hyponormal A, B* and normal C, D satisfy certain conditions, it is seen that the condition 0 set membership, variant σ(phi) is isolated in the set S = αβ + γδ:α set membership, variant σ(A), β set membership, variant σ(B), γ set membership, variant σ(C) and δ set membership, variant σ(D) is sufficient for phi−1(0) circled plus phi(B(H)) = B(H).