Spectra of T and T* when T is a bounded linear operator

  • Thread starter pyf
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  • #1
pyf
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Hi,

If T is a bounded linear operator on a Hilbert space, what can we say about the spectra of T and T* ([itex]\sigma(T)=\{\lambda:T-\lambda I[/itex] is not invertible})?
 

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  • #2
pyf
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I'm asking because they are equal for finite rank operators and I hope the relation is equally nice in infinite dimensions. :)
 
  • #3
dextercioby
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They're not related for arbitrary operators in [itex] \infty [/itex]-dimensional HS. If T is self-adjoint, the 2 spectra are equal.
 
  • #4
mathwonk
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Is it true at least for normal operators that the spectral radii are the same?
 
  • #5
Landau
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@mathwonk: I am not sure whether your question is meant to help the topic starter in the right direction, but since there hasn't been an answer yet, I'll give one:

Yes, then they are the same. A general theorem says that for a normal element x in a C*-algebra, its spectral radius equals its norm. Since x and x* have the same norm, they have the same spectral radius.
 

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