Let $A$ be a bounded self-adjoint operator on a Hilbert space. Assume the discrete spectrum of $A$ is empty. Let $B$ be a finite rank operator.
I have learned that the essential spectrum of $A+B$ and $A$ are the same, but what can be say about the discrete spectrum of $A+B$?
If we further assume $B$ is self-adjoint so it has $k =$ rank($B$) eigenvalues, then can we say that $A+B$ has at most $k$ eigenvalues?