spectral permanence theorem
For every element it makes sense to speak of the spectrum of relative to as well as the spectrum of relative to .
We provide here three results of increasing sophistication which relate both these spectrums, and . Any of the last two is usually refered to as the spectral permanence theorem.
- Let be as above. For every element we have
This first result is purely . It is a straightforward consequence of the fact that invertible elements in are also invertible in .
The other inclusion, , is not necessarily true. It is true, however, if one considers the boundary instead.
Theorem - Let be as above. For every element we have
Of course there may be only a finite number of bounded components or none.
Theorem - Let be as above. Then is obtained from by adjoining to it some (possibly none) bounded components of .
As an example, if is the unit circle, then can only possibly be the unit circle or the closed unit disk.
|Title||spectral permanence theorem|
|Date of creation||2013-03-22 17:29:50|
|Last modified on||2013-03-22 17:29:50|
|Last modified by||asteroid (17536)|