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.

中文翻译：

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无界线性算子的基本数值范围

我们引入了无界希尔伯特空间算子 $T$ 的基本数值范围 $W_{\!e}(T)$ 的概念，并研究了其基本性质，包括可能的等效特征和扰动结果。许多已知的有界情况的性质确实\emph{not} 延续到无界情况，并且出现了新的有趣现象，我们通过一些引人注目的例子来说明。基本数值范围 $W_{\!e}(T)$ 的一个关键特征是，当通过投影方法或 PDE 的域截断方法逼近 $T$ 时，它以统一和最小的方式捕获光谱污染。