Essential numerical range of elementary operators

Abstract:

Let $A= (A_{1},...,A_{p})$ and $B=(B_{1},...,B_{p})$ denote two $p$-tuples of operators with $A_{i}\in \mathcal L(H)$ and $B_{i}\in \mathcal L(K).$ Let $R_{2,A,B}$ denote the elementary operators defined on the Hilbert-Schmidt class $\mathcal C^{2}(H,K)$ by $R_{2,A,B}(X)=A_{1}XB_{1}+...+A_{p}XB_{p}.$ We show that \[ co\left [(W_{e}(A)\circ W(B))\cup (W(A)\circ W_{e}(B))\right ]\subseteq V_{e}(R_{2,A,B}).\] Here $V_{e}(.)$ is the essential numerical range, $W(.)$ is the joint numerical range and $W_{e}(.)$ is the joint essential numerical range.