Suppose I have a normal operator (not nec self-adjoint) $N$ on a separable Hilbert space. Suppose that I perturb by a compact operator $K$ to get (not nec normal)
$A=N+K$.
The essential spectra of $N$ and $A$ agree. My question is, does the addition of $K$ only cause the possibility of (at most countably many) eigenvalues to appear that can only accumulate at the essential spectrum?