A note on the norm of a basic elementary operator

Abstract

Let ${\cal L}(E)$ be the algebra of all bounded linear operators on a Banach space $E$. For $A,B\in{\cal L}(E)$, define the basic elementary operator $M_{A,B}$ by $M_{A,B}(X)=AXB$, ($X\in{\cal L}(E)$). If $\cal S$ is a symmetric norm ideal of ${\cal L}(E)$, we denote $M_{{\cal S},A,B}$ the restriction of $M_{A,B}$ to $\cal S$. In this paper, the norm equality $\|I+M_{{\cal S},A,B}\|=1+\|A\|\|B\|$ is studied. In particular, we give necessary and sufficient conditions on $A$ and $B$ for this equality to hold in the special case when $E$ is a Hilbert space and $\cal S$ is a Schatten $p$-ideal of ${\cal L}(E)$.