Norms of certain Jordan elementary operators

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. For A, B ∈ B(H), the Jordan elementary operator U A,B is defined by U A,B (X) = AXB + BX A, ∀X ∈ B(H). In this short note, we discuss the norm of U A,B . We show that if dimH = 2 and U A,B = A B , then either AB∗ or B∗ A is 0. We give some examples of Jordan elementary operators U A,B such that U A,B = A B but AB∗ = 0 and B∗ A = 0, which answer negatively a question posed by M. Boumazgour in [M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386–393].