On the range closure of an elementary operator Original Research Article

Let image denote the algebra of operators on a Hilbert image. Let image and image 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 image if and only if image is closed for some integer n greater-or-equal, slanted 1; (ii) if A, B, C and D are normal operators such that A commutes with C and B commutes with D, then image if and only if 0 set membership, variant iso σ(E).