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Let $G$ be normal operator with compact resolvent such that $\ker G$ is different from $\{0\}$.

Now Let $P$ be the orthogonal projection onto $\ker G$ and consider $G' = G + P$.

Please, I want an explication to the following question:

How $0$ belongs to resolvent set of $G'$?

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    I would have a complete definition about it.2013-12-22

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