Definition:Numerical Range

Definition

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $\norm {\, \cdot \,}$ be the inner product norm on $\struct {\HH, \innerprod \cdot \cdot}$.

Let $\struct {\map D T, T}$ be a densely-defined linear operator on $\HH$.

We define the numerical range $\map W T$ of $T$ by:

$\map W T = \set {\innerprod {T u} u : u \in \map D T, \, \norm u = 1}$

Sources

• 2018: David Edmunds and William Desmond Evans: Spectral Theory and Differential Operators (2nd ed.) ... (previous) ... (next) $3.2$: Numerical Range and Field of Regularity
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $16.1$: Eigenvalues of Self-Adjoint Operators