Let $G$ be an operator with compact resolvent on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$.

 Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$

 My question is: are these conditions sufficient to say that $0\in\rho(G_{0})$? If the answer is negative, the addition of the condition of $G$ is normal can gurantee that $0\in\rho(G_{0})$?