Spectrum of perturbed operator's

Question

Let $G$ be a normal operator with compact resolvent acting on a Hilbert space $H$ such that $\ker G \neq \{0\}$. If $P$ denotes the orthogonal projection onto $\ker G$, and if $\{\lambda_n\}$ are the eigenvalues of $G$, can we have $\lambda_{n}(G)\sim \lambda_{n}(G+P)$?

Answer 1

Hint. Using the appropriate Spectral Theorem for unbounded normal operators on Hilbert spaces (do your homework to find the most appropriate version) you should see that $G$ and $P$ are spectrally analyzed simultaneously, and hence $1$ is added to the spectrum of $G$ where it was $0$.

Answer 2

Hint. There is an orthogonal base in $H$ such that $G,P$ and $G+P$ are diagonal operators.