spectral permanence theorem

Let 𝒜 be a unital complex Banach algebraMathworldPlanetmath and 𝒜 a Banach subalgebra that contains the identityPlanetmathPlanetmath of 𝒜.

For every element x it makes sense to speak of the spectrum σ(x) of x relative to as well as the spectrum σ𝒜(x) of x relative to 𝒜.

We provide here three results of increasing sophistication which relate both these spectrums, σ(x) and σ𝒜(x). Any of the last two is usually refered to as the spectral permanence theorem.

- Let 𝒜 be as above. For every element x we have


This first result is purely . It is a straightforward consequence of the fact that invertible elements in are also invertiblePlanetmathPlanetmath in 𝒜.

The other inclusion, σ(x)σ𝒜(x), is not necessarily true. It is true, however, if one considers the boundary σ(x) instead.

Theorem - Let 𝒜 be as above. For every element x we have


Since the spectrum is a non-empty compact set in , one can decompose -σ𝒜(x) into its connected componentsMathworldPlanetmathPlanetmathPlanetmath, obtaining an unbounded componentMathworldPlanetmathPlanetmath Ω together with a sequence of boundedPlanetmathPlanetmathPlanetmathPlanetmath components Ω1,Ω2,,


Of course there may be only a finite number of bounded components or none.

Theorem - Let x𝒜 be as above. Then σ(x) is obtained from σ𝒜(x) by adjoining to it some (possibly none) bounded components of -σ𝒜(x).

As an example, if σ𝒜(x) is the unit circle, then σ(x) can only possibly be the unit circle or the closed unit disk.

Title spectral permanence theorem
Canonical name SpectralPermanenceTheorem
Date of creation 2013-03-22 17:29:50
Last modified on 2013-03-22 17:29:50
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 5
Author asteroid (17536)
Entry type Theorem
Classification msc 46H10
Classification msc 46H05