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Let
 be a unital complex Banach algebra and
 a Banach subalgebra that contains the identity of
 .
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.
Proposition - Let
 be as above. For every element
 we have
This first result is purely algebraic. 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
Since the spectrum is a non-empty compact set in
 , one can decompose
 into its connected components, obtaining an unbounded component
 together with a sequence of bounded components
 ,
Of course there may be only a finite number of bounded components or even 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.
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