Numerical range

Remarks

The numerical range is convex; this is the Toeplitz–Hausdorff theorem. The closure of $W(T)$ contains the spectrum of $T$. The numerical radius satisfies $\tfrac12 \norm{T} \le w(T) \le \norm{T}$, with the operator norm. 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.