Essential numerical range of elementary operators

Abstract

Let A = (A 1 ,..., Ap) and B = (B 1 ,..., Bp) denote two p-tuples of operators with A i ∈ £(H) and B i ∈ £(K). Let R 2,A,B denote the elementary operators defined on the Hilbert-Schmidt class C 2 (H, K) by R 2,A,B (X) = A 1 XB 1 +... + A p XB p . We show that co[(W e (A) o W(B)) U (W(A) o W e (B))] ⊂ 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.