A note on the essential numerical range of block diagonal operators

Abstract

In this note, we characterize the essential numerical range of a block diagonal operator T = ?i Ti in terms of the numerical ranges {W(Ti)}i of its components. Specifically, the essential numerical range of T is the convex hull of the limit superior of {W(Ti)}i. This characterization can be simplified further. In fact, we prove the existence of a decomposition of T for which the convex hull is not required.