Real numerical range

Definition

Real numerical range $W^{\mathbb{R}}(A)$ of a matrix $A$ of dimension $d$ is a subset of the complex plane defined as

$$W^{\mathbb{R}}(A)=\{z:z=⟨\psi |A|\psi ⟩,|\psi ⟩\in {\mathbb{R}}^d,⟨\psi |\psi ⟩=1\}.$$

Conditions for the generalized numerical range to be real

Fact

The classical numerical range satisfies $W(C{U}^{†}AU)\subset \mathbb{R}$ for all unitary $U$ if and only if at least one of $C$ and $A$ is scalar and their product is hermitian [1].

References

1. [1]M. Marcus and M. Sandy, “Conditions for the generalized numerical range to be real,” Linear algebra and its applications, vol. 71, pp. 219–239, 1985, [Online]. Available at: https://core.ac.uk/download/pdf/82435859.pdf.
   https://core.ac.uk/download/pdf/82435859.pdf