Norms of certain Jordan elementary operators (Q933488)

Let \(B(H)\) denote the algebra of all bounded linear operators acting on a Hilbert space \(H\). For \(A, B \in B(H)\), the Jordan elementary operator \(U_{A,B}\) is defined on \(B(H)\) by \(U_{A,B}(X) = AXB + BXA\). \textit{A.\,Blanco, M.\,Boumazgour} and \textit{T.\,J.\thinspace Ransford} [J.~Lond.\ Math.\ Soc., II.\ Ser.\ 70, No.\,2, 479--498 (2004; Zbl 1070.47025)] showed that \(\| U_{A,B}\| \geq \| A\| \,\| B\| \). This lower bound is the best known result to date. \textit{M.\,Boumazgour} [J.~Math.\ Anal.\ Appl.\ 342, No.\,1, 386--393 (2008; Zbl 1140.47023)] proved that if \(AB^* = B^*A = 0\), then \(\| U_{A,B}\| = \| A\| \,\| B\| \) and asked if \(\| U_{A,B}\| = \| A\| \,\| B\| \) implies that \(AB^*=B^*A = 0\)? In this note, the authors show that this is not true in general. They prove, however, that if \(\dim H=2\) and \(\| U_{A,B}\| =\| A\| \,\| B\| \), then \(AB^*=0\) or \(B^*A=0\).