A note on the essential numerical range of block diagonal operators

Luís Carvalho; Cristina Diogo; Sérgio Mendes; Helena Soares

paper

A note on the essential numerical range of block diagonal operators

arXiv:2212.11659

Abstract

In this note we characterize the essential numerical range of a block diagonal o\-pe\-ra\-tor $T = ⨁_{i} T_{i}$ in terms of the numerical ranges ${W (T_{i})}_{i}$ of its components. Specifically, the essential numerical range of $T$ is the convex hull of the limit superior of ${W (T_{i})}_{i}$ . 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.