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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].


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


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) = $, when$W_e(T) $

- 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) $, then$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) $,$W_e(T) D$and$$ intersects $\partial D$ at infinitely many points, then $W(T) = $ [5].


  1. [1]J. G. Stampfli and J. P. Williams, “Growth conditions and the numerical range in a Banach algebra,” Tohoku Mathematical Journal, Second Series, vol. 20, no. 4, pp. 417–424, 1968, [Online]. Available at:
  2. [2]P. A. Fillmore, J. G. Stampfli, and J. P. Williams, “On the essential numerical range, the essential spectrum, and a problem of Halmos,” Acta Sci. Math.(Szeged), vol. 33, no. 197, pp. 179–192, 1972, [Online]. Available at:
  3. [3]S. Bogli and M. Marletta, “Essential numerical ranges for linear operator pencils,” arXiv preprint arXiv:1909.01301, vol. 0, 2019, [Online]. Available at:
  4. [4]S. Bogli, M. Marletta, and C. Tretter, “The essential numerical range for unbounded linear operators,” Journal of Functional Analysis, vol. 0, p. 108509, 2020, [Online]. Available at:
  5. [5]B. Lins, “Numerical ranges encircled by analytic curves,” arXiv preprint arXiv:2003.05347, vol. 0, 2020, [Online]. Available at: