On the range closure of an elementary operator

Let B(H) denote the algebra of operators on a Hilbert H. Let Delta(AB) is an element of B(B(H)) and E is an element of B(B(H)) denote the elementary operators Delta(AB)(X) = AXB - X and E(X) = AXB - CXD. We answer two questions posed by Turnsek [Mh. Math. 132 (2001) 349-354] to prove that: (i) if A, B are contractions, then B(H) = Delta(-1)(AB) (0) circle times Delta(AB) (B(H)) if and only if Delta(n)(AB)(B(H)) is closed for some integer n >= 1; (ii) if A, B, C and D are normal operators such that A commutes with C and B commutes with D, then B(H) = E-1(0) circle plus E(B(H)) if and only if 0 is an element of iso sigma (E). (c) 2005 Elsevier Inc. All rights reserved.