An equality condition for the norm of an elementary operator

Abstract

Let $\cal B(H)$ be the algebra of all bounded linear operators on a complex Hilbert space $H$. The elementary operator $R_{\mathbf{A}, \mathbf{B}}$ induced by two $n$-tuples $\mathbf{A}=(A_{1}, \dots, A_n)$ and $\mathbf{B}=(B_{1}, \dots, B_{n})$ of elements of $\mathcal{B(H)}$ is defined by $$R_{\mathbf{A}, \mathbf{B}}(X)=\sum_{i=1}^nA_iXB_i\, (X\in\cal B(H)).$$
The aim of this paper is to give necessary and sufficient conditions under which the supremum $\sup\big\{\|R_{\mathbf{A}, \mathbf{B}}(X)\|:X\in \mathcal B(H),\,\|X\|=1,\, \hbox{rank} X=1\big\}$ attains its optimal value $\frac{1}{2}\sum_{i=1}^{n} \left(\| A_{i}\|^{2}+\| B_{i}\|^{2}\right)$.