spectral mapping theorem
Let be a unital -algebra (http://planetmath.org/CAlgebra). Let be a normal element in and be its spectrum.
Spectral Mapping Theorem - Let be as above. Let . Then
Proof : Since and are isomorphic we must have
where denotes the spectrum of relative to the subalgebra .
By the spectral invariance theorem we have . Hence
Thus, we only have to prove that .
is defined on so is precisely the image of .
Equivalently, is not invertible if and only if has a zero, i.e. for some .
The previous statement can be reformulated as: if and only if is in the image of .
We conclude that , and this proves the theorem.
|Title||spectral mapping theorem|
|Date of creation||2013-03-22 17:30:08|
|Last modified on||2013-03-22 17:30:08|
|Last modified by||asteroid (17536)|