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 $0\in\rho(G_{0})$?

Let $G$ be an operator with compact resolvent on a Hilbert space $H$ such that $\ker G \neq \{0\}$. Further 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, does the additional condition that $G$ is normal guarantee that $0\in\rho(G_{0})$?