On the range closure of an elementary operator

Let B(H) denote the algebra of operators on a Hilbert H. Let ΔAB∈B(B(H)) and E∈B(B(H)) denote the elementary operators ΔAB(X)=AXB−X and E(X)=AXB−CXD. We answer two questions posed by Turnšek [Mh. Math. 132 (2001) 349–354] to prove that: (i) if A, B are contractions, then B(H)=ΔAB-1(0)⊕ΔAB(B(H)) if and only if ΔABn(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)⊕E(B(H)) if and only if 0∈isoσ(E).