Norms of elementary operators

Abstract:

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 $\sup \{\parallel \!\!\sum _{i=1}^n A_iXB_i\!\!\parallel ~: X\in \mathcal {B(H)}, \parallel \!\!X\!\!\parallel \ \leq 1\}=\sup \{\parallel \!\!\sum _{i=1}^n A_iUB_i\!\!\parallel \ : UU^*=U^*U=I, U\in {\mathcal {B(H)}}\}.$ Moreover, we prove that there exists an operator $X_0$ with $\parallel \!\! X_0\!\!\parallel \ =1$ such that $\parallel \!\!\sum _{i=1}^n A_iX_0B_i\!\!\parallel \ =\sup \{\parallel \!\!\sum _{i=1}^n A_iXB_i\!\!\parallel \ : X\in {\mathcal {B(H)}}, \parallel \!\!X\!\!\parallel \ \leq 1\}$ if and only if there exists a unitary $U_0\in \mathcal {B(H)}$ such that $\parallel \!\!\sum _{i=1}^n A_iU_0B_i\!\!\parallel \ =$ $\sup \{\parallel \!\!\sum _{i=1}^n A_iXB_i\!\!\parallel \ : X\in {\mathcal {B(H)}}, \parallel \!\!X\!\!\parallel \ \leq 1\}.$