# spectral values classification

Spectral points classificationFernando Sanz Gamiz

###### Definition 1.

Let $X$ a topological vector space^{} and $A:X\supset {D}_{A}\u27f6X$ a linear transformation with
domain ${D}_{A}$. Depending on the properties of^{1}^{1}the notation $(\lambda -A)$ is to be
understood as $\lambda I-A$ with $I$ the identity transformation and $R(\lambda -A)$ is the range
of $(\lambda -A)$ $(\lambda -A)$ the following definitions apply:

${(\lambda -A)}^{-1}$ | Boundness of ${(\lambda -A)}^{-1}$ | $R(\lambda -A)$ | Set to which $\lambda $ belongs |
---|---|---|---|

exists | bounded | dense in X | resolvent set $\rho (A)$ |

exists | unbounded | dense in X | continuous spectrum $C\sigma (A)$ |

exists | bounded or unbounded in X | not dense in X | residual spectrum $R\sigma (A)$ |

not exists | dense or not dense in X | puntual spectrum $P\sigma (A)$ |

###### Remark 1.

It is obvious that, if $F$ is the field of possible values for $\lambda $ (usually $F=\u2102$ or
$F=\mathbb{R}$) then $F=\rho (A)\cup C\sigma (A)\cup R\sigma (A)\cup P\sigma (A)$, that is, these
definitions cover all the possibilities for $\lambda $. The complement of the resolvent set is called *spectrum* of the operator A, i.e., $\sigma (A)=C\sigma (A)\cup R\sigma (A)\cup P\sigma (A)$

###### Remark 2.

In the finite dimensional case if ${(\lambda -A)}^{-1}$ exists it must be bounded, since all finite dimensional linear mappings are bounded. This existence also implies that the range of $(\lambda -A)$ must be the whole X. So, in the finite dimensional case the only spectral values we can encounter are point spectrum values (eigenvalues).

Title | spectral values classification |

Canonical name | SpectralValuesClassification |

Date of creation | 2013-03-22 18:52:01 |

Last modified on | 2013-03-22 18:52:01 |

Owner | fernsanz (8869) |

Last modified by | fernsanz (8869) |

Numerical id | 5 |

Author | fernsanz (8869) |

Entry type | Definition |

Classification | msc 15A18 |

Synonym | eigenvalues |

Synonym | spectrum |

Related topic | Eigenvalue |

Related topic | SpectrumOfAMuI |

Related topic | InvertibleLinearTransformation |

Defines | spectrum |

Defines | point spectrum |

Defines | residual spectrum |

Defines | continuous spectrum |

Defines | resolvent set |

Defines | eigenvalues |

Defines | puntual spectrum |

Defines | point spectral value |

Defines | residual spectral value |

Defines | continuous spectral value |

Defines | resolvent set value |