Norms of elementary operators

Let $A_i$ and $B_i$, $1\leq i\leq n$, be bounded linear operators acting on a separable Hilbert space $\mathcal H$. In this note, we prove that $\sup\{\parallel\sum_{i=1}^n A_iXB_i\parallel~: X\in \mathcal{B(H)}, \parallelX... ...{\parallel\sum_{i=1}^n A_iUB_i\parallel : UU^*=U^*U=I, U\in {\mathcal{B(H)}}\}.$ Moreover, we prove that there exists an operator $X_0$ with $\parallel X_0\parallel =1$ such that $\parallel\sum_{i=1}^n A_iX_0B_i\parallel =\sup\{\parallel\sum_{i=1}^n A_iXB_i\parallel : X\in {\mathcal{B(H)}}, \parallelX\parallel \leq 1\}$ if and only if there exists a unitary $U_0\in \mathcal{B(H)}$ such that $\parallel\sum_{i=1}^n A_iU_0B_i\parallel =$ $\sup\{\parallel\sum_{i=1}^n A_iXB_i\parallel : X\in {\mathcal{B(H)}}, \parallelX\parallel \leq 1\}.$