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 ∈ iso σ(E).