A counter example for adjoint of unbounded operators

Question

I need a counter example for $(A+B)^*=A^*+B^*$, where $A$ and $B$ are unbounded operators on Hilbert space and $^*$ denotes the adjoint.

Answer 1

Take $A=-B$. Then $(A+B)^*$ is the zero operator defined on the entire space, but $A^*+B^*$ is the zero operator restricted to $\mathcal{D}(A^*)=\mathcal{D}(B^*)$.

However we do have $A^*+B^*\subset (A+B)^*$.

$A^*+B^*=(A+B)^*$ holds for example if $A$ or $B$ is bounded.