Sum of a self adjoint operator and compact operator

Question

Suppose $A$ is a self adjoint operator and $B$ is a compact operator.

Can we say that $A+B$ is a self-adjoint operator?

Answer 1

No, take $A = 0$ and $B$ any compact operator which is not self-adjoint. For example $B = i \cdot \mathrm{id}$ on $M_n(\mathbb C)$.