Lower bound for the norm of the resolvent

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In summary, the conversation discusses the norm of the resolvent for self-adjoint operators in the context of quantum mechanics. The footnote on page 713 states that the norm is greater than or equal to the inverse of the distance of z from the closest eigenvalue of A. The conversation also asks for suggestions on how to prove this, including links or answers. Additionally, it suggests looking at the effects of operators A-zI and (A-zI)^{-1} on eigenvectors in a finite-dimensional space.
  • #1
alle.fabbri
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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
[tex]
\|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}
[/tex]
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...
 
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  • #2
If you are in a finite-dimensional space, and if [tex]\lambda[/tex] is the closest to [tex]z[/tex] eigenvalue, look what operators [tex]A- z I[/tex]
and [tex](A- z I)^{-1}[/tex] do with the corresponding eigenvector.
 

1. What is the "Lower bound for the norm of the resolvent"?

The lower bound for the norm of the resolvent is a mathematical concept used in functional analysis to determine the minimum distance between a given point and the spectrum of a linear operator.

2. Why is the lower bound for the norm of the resolvent important?

The lower bound for the norm of the resolvent is important because it provides information about the behavior and properties of a linear operator. It can also be used to prove the convergence of certain numerical methods.

3. How is the lower bound for the norm of the resolvent calculated?

The lower bound for the norm of the resolvent is calculated using the spectral properties of a linear operator, such as its spectrum and eigenvalues. It can also be calculated using the numerical range of the operator.

4. Can the lower bound for the norm of the resolvent be negative?

No, the lower bound for the norm of the resolvent cannot be negative. It is always a positive value that represents the minimum distance between a point and the spectrum of a linear operator.

5. How is the lower bound for the norm of the resolvent used in practical applications?

The lower bound for the norm of the resolvent is used in various fields, such as physics, engineering, and mathematics. It is particularly important in the analysis of differential equations and the design of control systems.

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