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Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex plane that separates $\lambda$ from the rest of the spectrum of $A$. Then $ - \frac{1}{2 \pi i} \int_{C_\lambda}{(A-zI)^{-1}dz} $ is the projection of $H$ onto the eigenspace of $\lambda$.

Where can I find a proof of this theorem? I am looking for an introductory text that maybe also sheds some light on the theory around this claim, i.e. the definition of the contour integral and the relation of this integral to the measurable functional calculus and the spectral measure of $A$.

EDIT: Assume that $A$ is normal or self-adjoint.

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    A last comment: you should ping users using `@username` if you want them to be notified by your comments. I only saw your follow-up by accident, hence the late response.2012-06-15

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