Question about Spectral Theory

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The discussion centers on spectral theory, specifically regarding a self-adjoint operator L that satisfies the condition ≥ 0. It is established that the spectrum σ(L) is contained within the interval [0, ∞). The participants clarify that the self-adjoint property leads to the conclusion that the eigenvalues, represented by λ, must be non-negative due to the non-negativity of the inner product and the norm of the function f. This understanding is crucial for further exploration of spectral properties in functional analysis.

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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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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)?
 
Yes, I emailed my professor and it was supposed to be ">=" instead of "=".

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

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