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# Essential numerical range

The essential numerical range $W_e(T)$ was introduced by Stampfli and Williams in [1] for a bounded linear operator $T$ in a Hilbert space $H$ as the closure of the numerical range of the image of $T$ in the Calkin algebra. Various equivalent characterizations were established, for example, in [2], along with applicatations, for example, in [3].

## Definition

For a linear operator T with domain $\mathcal{D}(T) \subset H$ we define the essential numerical range of $T$ by $$W_e(T ) = \{\lambda \in \mathbf{C} : \exists (x_n)_{n \in \mathbf{N} }\subset \mathcal{D}(T ) \,\,\, \text{with} \,\,\, ||x_n|| = 1, x_n \rightarrow 0, \braket{Tx_n}{x_n} \rightarrow \lambda \} .$$

### Properties

For any $z \in \mathbb{C}$ and let $$\sigma_e(T) = \{ \lambda \in \mathrm{C} : \exists (x_n)_{n \in \mathbb{N}} \subset \mathcal{D}(T ) \,\,\, \text{with} \,\,\, ||xn|| = 1, (x_n) \rightarrow 0, ||(T- \lambda)x_n || \rightarrow 0 \}$$ then we have [4]

- $W_e(zT) = zW_e(T)$

- $W_e(T+z) = W_e(T) + z$

- $W_e(T)$ is closed and convex

- $\text{conv} \, \sigma_e(T) \subset W_e(T )$

### Relation between numerical range $W(T)$ and essential numerical range $W_e(T)$

Let $W_e(T)$ will be the essential numerical range of $T$ whereas $W(T)$ will be the numerical range of $T$ and $\overline{W(T)}$ the closure of $W(T)$. Then we have the following relations:

- $W_e(T) \subset \overline{W(T)}$

- If $\overline{W(T)}$ is a line or a strip or $W(T) = \mathbb{C}$, when $W_e(T) \neq \emptyset$

- If $W(T)$ is a line, then so is $W_e(T)$ and thus $W_e(T) = W(T)$

- If $W(T) = \mathbb{C}$ if and only if $W_e(T) = \mathbb{C}$

- If $\overline{W(T)}$ is a strip, then $W_e(T)$ is a line or a strip

- If $\overline{W(T)}$ is a half-plane and $W_e(T) \neq \emptyset$, then $W_e(T)$ is a half-plane

- If $D$ is a bounded open convex subset of $\mathbb{C}$ with a regular analytic boundary curve $\partial D$, $W(T) \subseteq \overline{D}$, $W_e(T) \subset D$ and $\overline{W(T)}$ intersects $\partial D$ at infinitely many points, then $W(T) = \overline{D}$ [5].

1. JG Stampfli, JP Williams, 1968. Growth conditions and the numerical range in a Banach algebra. Tohoku Mathematical Journal, Second Series, 20, Mathematical Institute, Tohoku University, pp.417–424.
2. PA Fillmore, JG Stampfli, James P Williams, 1972. On the essential numerical range, the essential spectrum, and a problem of Halmos. Acta Sci. Math.(Szeged), 33, pp.179–192.
3. Sabine Bogli, Marco Marletta, 2019. Essential numerical ranges for linear operator pencils. arXiv preprint arXiv:1909.01301.
4. Sabine Bogli, Marco Marletta, Christiane Tretter, 2020. The essential numerical range for unbounded linear operators. Journal of Functional Analysis, Elsevier, pp.108509.
5. Brian Lins, 2020. Numerical ranges encircled by analytic curves. arXiv preprint arXiv:2003.05347.