Let $G$ be an operator on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$.
Let also $P$ be the orthogonal projection onto $\ker G$, $G_1$ the restriction of $G$ on $\ker P$ and
$G_2$ the restriction of $G$ on $\operatorname{Im} P$.
My question is: can I say that $\sigma(G) = \sigma(G_{1})\cup\sigma(G_{2})$?