# Simple isolated eigenvalue and pole of the resolvent

Let $T$ be bounded linear operator on some complex Banach space, and $\lambda$ an eigenvalue of $T$ which is isolated in its spectrum, and such that $\bigcup_{n\ge 1} N((T- \lambda I)^n)$ is one-dimensionnal.

How can we prove that $\lambda$ is a simple pole of the resolvent of $T$ ?

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Dear Ahriman, Can you explain what you mean by $N((T-\lambda)^n)$? Thanks! Regards, – Matt E Nov 27 '11 at 13:17
This denotes the kernel of the operator $(T-\lambda I)^n$. – Ahriman Nov 27 '11 at 16:25