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

[pyf]

Hi,

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

[pyf]

I'm asking because they are equal for finite rank operators and I hope the relation is equally nice in infinite dimensions. :)

[dextercioby]

They're not related for arbitrary operators in \infty -dimensional HS. If T is self-adjoint, the 2 spectra are equal.

[mathwonk]

Is it true at least for normal operators that the spectral radii are the same?

[Landau]

@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.