# spectral permanence theorem

Let $\mathcal{A}$ be a unital complex Banach algebra^{} and $\mathcal{B}\subseteq \mathcal{A}$ a Banach subalgebra that contains the identity^{} of $\mathcal{A}$.

For every element $x\in \mathcal{B}$ it makes sense to speak of the spectrum ${\sigma}_{\mathcal{B}}(x)$ of $x$ relative to $\mathcal{B}$ as well as the spectrum ${\sigma}_{\mathcal{A}}(x)$ of $x$ relative to $\mathcal{A}$.

We provide here three results of increasing sophistication which relate both these spectrums, ${\sigma}_{\mathcal{B}}(x)$ and ${\sigma}_{\mathcal{A}}(x)$. Any of the last two is usually refered to as the spectral permanence theorem.

- Let $\mathcal{B}\subseteq \mathcal{A}$ be as above. For every element $x\in \mathcal{B}$ we have

$${\sigma}_{\mathcal{A}}(x)\subseteq {\sigma}_{\mathcal{B}}(x).$$ |

This first result is purely . It is a straightforward consequence of the fact that invertible elements in $\mathcal{B}$ are also invertible^{} in $\mathcal{A}$.

The other inclusion, ${\sigma}_{\mathcal{B}}(x)\subseteq {\sigma}_{\mathcal{A}}(x)$, is not necessarily true. It is true, however, if one considers the boundary $\partial {\sigma}_{\mathcal{B}}(x)$ instead.

Theorem - Let $\mathcal{B}\subseteq \mathcal{A}$ be as above. For every element $x\in \mathcal{B}$ we have

$$\partial {\sigma}_{\mathcal{B}}(x)\subseteq {\sigma}_{\mathcal{A}}(x).$$ |

Since the spectrum is a non-empty compact set in $\u2102$, one can decompose $\u2102-{\sigma}_{\mathcal{A}}(x)$ into its connected components^{}, obtaining an unbounded component^{} ${\mathrm{\Omega}}_{\mathrm{\infty}}$ together with a sequence of bounded^{} components ${\mathrm{\Omega}}_{1},{\mathrm{\Omega}}_{2},\mathrm{\dots}$,

$$\u2102-{\sigma}_{\mathcal{A}}(x)={\mathrm{\Omega}}_{\mathrm{\infty}}\cup {\mathrm{\Omega}}_{1}\cup {\mathrm{\Omega}}_{2}\cup \mathrm{\cdots}$$ |

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

Theorem - Let $x\in \mathcal{B}\subseteq \mathcal{A}$ be as above. Then ${\sigma}_{\mathcal{B}}(x)$ is obtained from ${\sigma}_{\mathcal{A}}(x)$ by adjoining to it some (possibly none) bounded components of $\u2102-{\sigma}_{\mathcal{A}}(x)$.

As an example, if ${\sigma}_{\mathcal{A}}(x)$ is the unit circle, then ${\sigma}_{\mathcal{B}}(x)$ 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 |