I have found this:
Let $A,B:X\rightarrow X$ linear bounded operators. If $A$ differs from $B$ by a compact operator, then we have that $A$ and $B$ have spectrally equivalence modulo a compact operator. But what this means?
I think we do not have that if $A=B+C$, where $C$ is compact, then $\sigma(A)=\sigma(B)$. Think about $A=B=I$. So, I don't understand what means spectrally modulo a compact operator.