Additivity of the Field of Values

Question

If $A \in M_n(\mathbb{C})$, then the field (of values), or numerical range of A, is the compact, convex subset of the complex-plane defined by $$ F(A)= \{z^* A z \mid z^*z = 1 \}. $$

It is well-known that the field is sub-additive in the sense that \begin{equation} F(A+B) \subseteq F(A) + F(B), \tag{1}\label{fov} \end{equation} where addition-symbol on the right denotes Minkowksi addition.

Question: Is there a characterization known for when equality holds in \eqref{fov}?

Notice that if $A = UDU^*$ and $B= U \hat{D} U^*$, with $U$ unitary and $D$, $\hat{D} $ diagonal, then it is easily shown that equality holds.

EDIT: But this condition is not necessary as equality also holds when $B = cA$, $c \in \mathbb{C}$.

Answer 1

This is just a partial answer, but maybe an important case.

If $A$ is hermitian, $F(A)$ is the interval $[\lambda_{\text{min}}(A), \lambda_{\text{max}}(A)]$, where $\lambda_{\text{min}}(A)$ and $\lambda_{\text{max}}(A)$ are the minimum and maximum eigenvalues of $A$. Thus if $A$ and $B$ are both hermitian, $F(A+B) = F(A) + F(B)$ means $\lambda_{\text{min}}(A+B) = \lambda_{\text{min}}(A)+\lambda_{\text{min}}(B)$ and similarly for $\lambda_{\text{max}}$. That will happen if and only if $A$ and $B$ share a common eigenvector for their minimum eigenvalues and share a common eigenvector for their maximum eigenvalues.

EDIT: It is not true that equality holds when $B = cA$, $c \in \mathbb C$. For example, suppose $A = \pmatrix{0 & 0\cr 0 & 1\cr}$ and $B = i A$. Then $F(A) = [0,1]$, $F(B) = i[0,1]$, and $F(A)+F(B)$ is the convex hull of $0$, $1$, $i$ and $1+i$, but $F(A+B)$ is only the convex hull of $0$ and $1+i$.