A note concerning the numerical range of a basic elementary operator

Abstract

Let $\mathcal{B}\left(H\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $H$, and let $\mathcal{S}$ be a norm ideal in $\mathcal{B}\left(H\right)$. For $A,B\in \mathcal{B}\left(H\right)$, define the elementary operator $M_{\mathcal{S},A,B}$ on $\mathcal{S}$ by $M_{\mathcal{S},A,B}\left(X\right)=AXB$ ($X\in \mathcal{S}$). The aim of this paper is to give necessary and sufficient conditions under which the equality $V\left(M_{\mathcal{S},A,B}\right)=\overline{co}\left(W\right(A\left)W\right(B\left)\right)$ holds. Here $V\left(T\right)$ and $W\left(T\right)$ denote the algebraic numerical range and spatial numerical range of an operator $T$, respectively, and $\overline{co}(\Omega )$ denotes the closed convex hull of a subset $\Omega \subseteq \mathbb{C}$.