A note on the essential numerical range of block diagonal operators

TL;DR

This paper characterizes the essential numerical range of block diagonal operators using the numerical ranges of their components, providing a simplified and more precise understanding of their spectral properties.

Contribution

It offers a new characterization of the essential numerical range for block diagonal operators, linking it directly to the limit superior of component numerical ranges.

Findings

Essential numerical range equals the convex hull of the limit superior of component ranges.

A decomposition exists where the convex hull is unnecessary for the characterization.

Simplifies understanding of spectral properties of block diagonal operators.

Abstract

In this note we characterize the essential numerical range of a block diagonal o\-pe\-ra\-tor $T=\bigoplus_i T_i$ 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.