Lower bound for the norm of the resolvent

[alle.fabbri]

Hi all!
I hope this is the right section to post such a question...
I'm studying the theory of resolvent from the QM books by A. Messiah and I read in a footnote (page 713) that the norm of the resolvent satisfies
<br /> \|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}<br />
where the equality holds for self-adjoint operators and "dist" is the distance of z from the closest eigenvalue of A. Any idea of how to prove this? Links are good as well as answers...

Thanks to anyone who will answer...

 

[Hawkeye18]

If you are in a finite-dimensional space, and if \lambda is the closest to z eigenvalue, look what operators A- z I
and (A- z I)^{-1} do with the corresponding eigenvector.