On the Normality of the Sum of Two Normal Operators

Question

It is well kown that

Theorem: Let $A$ and $B$ be two normal operators. If $A$ commutes with $B$, then $A + B$ is normal.

Indeed the proof follows by using the Fuglede theorem since the commutativity of $A$ and $B$ implies the commutativity of $A$ and $B^*$.

The converse of this theorem is not always true. I hope to construct a counterexample.

Answer 1

Take, for instance,$$A=\begin{pmatrix}1&0\\0&2\end{pmatrix}\text{ and }B=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$These matrices are normal and they don't commute, but $A+B$ is still normal. Actually, it's symmetric.