spectral mapping theorem
Let be a unital -algebra (http://planetmath.org/CAlgebra). Let be a normal element in and be its spectrum.
The continuous functional calculus provides a -isomorphism
between the -algebra of complex valued continuous functions on and the -subalgebra generated by and the identity of .
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 .
Let . The function is invertible if and only if has no zeros.
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 |
---|---|
Canonical name | SpectralMappingTheorem |
Date of creation | 2013-03-22 17:30:08 |
Last modified on | 2013-03-22 17:30:08 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 4 |
Author | asteroid (17536) |
Entry type | Theorem |
Classification | msc 46L05 |
Classification | msc 47A60 |
Classification | msc 46H30 |