Norm equality for a basic elementary operator

Let L(H) be the algebra of bounded linear operators on a Hilbert space H. For A,B ∈ L(H), define the elementary operator MA,B by MA,B(X) = AXB (X ∈ L(H)). We give necessary and sufficient conditions for any pair of operators A and B to satisfy the equation I +MA,B = 1 + A B , where I is the identity operator on H.  2003 Published by Elsevier Inc.