The closure of the range of an elementary operator

Let B(H) denote the algebra of operators on a Hilbert space H, and let phi epsilon B(B(H)) be the elementary operator defined by phi(X) = AXB + CXD. A necessary condition for phi(-1) (0) circle plus phi 4) (B (H)) = B (H) is that 0 is an isolated point of the spectrum sigma (phi) of phi. We prove a sufficient condition for phi(-1) (0) circle 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 E sigma (phi) is isolated in the set S = {alphabeta + gammadelta : alpha epsilon sigma (A) beta epsilon sigma (B), gamma epsilon sigma (C) and delta epsilon sigma (D)} is sufficient for phi(-1) (0) circle plus phi (B(H)) = B(H). (C) 2004 Elsevier Inc. All rights reserved.