---
tag: Y730CQ2J1F
kind: definition
name: Numerical range
msc: [47A12]
defines: XWXZ0DGW3Z
uses: [8TV0DJBBAY, HM2HHE3J51, 03TM135G0N, BMKATJ802A, 06EKW1AKXF]
sources:
  - url: https://en.wikipedia.org/wiki/Numerical_range
---

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$, and let $T \colon H \to H$ be a [[continuous linear map|03TM135G0N]].
The *numerical range* of $T$ is the set
$$W(T) = \set{\ip{Tx}{x} : x \in H, \ \norm{x} = 1} \subseteq \C.$$
The *numerical radius* of $T$ is $w(T) = \sup \set{\abs{z} : z \in W(T)}$.

## Remarks

The numerical range is convex; this is the [[Toeplitz–Hausdorff theorem|8TV0DJBBAY]].
The closure of $W(T)$ contains the [[spectrum|BMKATJ802A]] of $T$.
The numerical radius satisfies $\tfrac12 \norm{T} \le w(T) \le \norm{T}$, with the [[operator norm|06EKW1AKXF]].
The operator $T$ is self-adjoint if and only if $W(T) \subseteq \R$.

Example: let $T = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ on $\C^2$.
For $x = (x_1, x_2)$ with $\abs{x_1}^2 + \abs{x_2}^2 = 1$, $\ip{Tx}{x} = x_2 \overline{x_1}$, whose absolute value is at most $\tfrac12$ and takes every value with $\abs{z} \le \tfrac12$.
So $W(T)$ is the closed disc of radius $\tfrac12$, and $w(T) = \tfrac12 \norm{T}$.

Non-example: for the same $T$ and $x = (1, 1)$, $\ip{Tx}{x} = 1$, which is not in $W(T)$; the vector $x$ is not a unit vector.