# The essential numerical range for unbounded linear operators

## TL;DR

This paper introduces the essential numerical range for unbounded operators on Hilbert spaces, exploring its properties, differences from the bounded case, and its role in spectral pollution during approximation methods.

## Contribution

It defines and studies the essential numerical range for unbounded operators, revealing new phenomena and its application in spectral pollution analysis.

## Key findings

- Essential numerical range captures spectral pollution.
- Many properties from bounded case do not extend to unbounded operators.
- Provides new characterizations and perturbation results for $W_e(T)$.

## Abstract

We introduce the concept of essential numerical range $W_{\!e}(T)$ for unbounded Hilbert space operators $T$ and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do \emph{not} carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range $W_{\!e}(T)$ is that it captures spectral pollution in a unified and minimal way when approximating $T$ by projection methods or domain truncation methods for PDEs.