Question about Spectral Theory

[mathmajor314]

Let L be a self-adjoint operator satisfying <Lf,f>=0. Show that \sigma(L)\subseteq[0,\infty).

I know that L being self-adjoint implies that <Lf,f>=<\lambdaf,f>=\lambda<f,f>=\lambdanorm(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!

 

[Dick]

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)?

 

[mathmajor314]

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

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