Question about Spectral Theory

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In summary, the conversation discusses a self-adjoint operator L satisfying <Lf,f> >= 0 and its implication that <Lf,f> is equal to lambda times the norm of f. This leads to the conclusion that lambda must be a non-negative number, and therefore <Lf,f> can only take on values in the range of [0, infinity).
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mathmajor314
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Let L be a self-adjoint operator satisfying <Lf,f>=0. Show that [tex]\sigma[/tex](L)[tex]\subseteq[/tex][0,[tex]\infty[/tex]).

I know that L being self-adjoint implies that <Lf,f>=<[tex]\lambda[/tex]f,f>=[tex]\lambda[/tex]<f,f>=[tex]\lambda[/tex]norm(f).

And <Lf,f>=<f,L*f>=<f,Lf>. I'm not sure where to go from here though.

Thank you in advance for any help!
 
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  • #2
I think you mean '<Lf,f> >= 0', not '<Lf,f> = 0'. If that's equal to lambda*norm(f), what does that tell you about lambda? What kind of number is norm(f)?
 
  • #3
Yes, I emailed my professor and it was supposed to be ">=" instead of "=".

After changing his typo, I figured it out. Thanks!
 

1. What is Spectral Theory?

Spectral Theory is a branch of mathematics that deals with the properties of linear operators on vector spaces. It studies the relationship between the eigenvalues and eigenvectors of a linear operator, and how these can be used to analyze the behavior of the operator.

2. Why is Spectral Theory important?

Spectral Theory is important because it provides a powerful tool for understanding the properties of linear operators and their associated systems. It has applications in various fields such as physics, engineering, and computer science.

3. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts in linear algebra that are important in Spectral Theory. Eigenvalues are the numbers that represent the scaling factor of an eigenvector when it is multiplied by a linear operator. Eigenvectors are the vectors that remain in the same direction after being multiplied by a linear operator.

4. How is Spectral Theory used in real-world applications?

Spectral Theory has various applications in real-world problems, such as analyzing the stability of physical systems, understanding the behavior of electrical circuits, and even in image processing and data analysis. It is also used in quantum mechanics to study the properties of atoms and molecules.

5. What are some common techniques used in Spectral Theory?

Some common techniques used in Spectral Theory include diagonalization, which involves finding a basis of eigenvectors for a given linear operator, and the spectral theorem, which states that a self-adjoint operator can be represented by a diagonal matrix with real eigenvalues. Another technique is the use of the characteristic polynomial to find the eigenvalues of a given operator.

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