# resolvent function is analytic

Theorem - Let $\mathcal{A}$ be a complex Banach algebra^{} with identity element^{} $e$. Let $x\in \mathcal{A}$ and $\sigma (x)$ denote its spectrum.

Then, the resolvent function (http://planetmath.org/ResolventMatrix) ${R}_{x}:\u2102-\sigma (x)\u27f6\mathcal{A}$ defined by ${R}_{x}(\lambda )={(x-\lambda e)}^{-1}$ is analytic (http://planetmath.org/BanachSpaceValuedAnalyticFunctions).

Moreover, for each ${\lambda}_{0}\in \u2102-\sigma (x)$ it has the power series

${R}_{x}(\lambda )={\displaystyle \sum _{n=0}^{\mathrm{\infty}}}{R}_{x}{({\lambda}_{0})}^{n+1}{(\lambda -{\lambda}_{0})}^{n}$ | (1) |

where the series converges absolutely for each $\lambda $ in the open disk centered in ${\lambda}_{0}$ given by

$$ | (2) |

Proof : Analyticity is defined for functions whose domain is open.

Thus, we start by proving that $\u2102-\sigma (x)$ is an open set in $\u2102$. To do so it is enough to prove that for every ${\lambda}_{0}\in \u2102-\sigma (x)$ the open disk defined by (2) above is contained in $\u2102-\sigma (x)$.

Let ${\lambda}_{0}\in \u2102-\sigma (x)$ and $\lambda $ be such that

$$ |

Then $$ and by the Neumann series (http://planetmath.org/NeumannSeriesInBanachAlgebras) $e-(\lambda -{\lambda}_{0}){R}_{x}({\lambda}_{0})$ is invertible.

Since ${\lambda}_{0}\notin \sigma (x)$ it follows that $(x-{\lambda}_{0}e)$ is invertible.

Hence, from the equality

$x-\lambda e=x-{\lambda}_{0}e-(\lambda -{\lambda}_{0})e=(x-{\lambda}_{0}e)\cdot [e-(\lambda -{\lambda}_{0}){R}_{x}({\lambda}_{0})]$ | (3) |

we conclude that $x-\lambda e$ is also invertible, i.e. $\lambda \in \u2102-\sigma (x)$. Thus $\u2102-\sigma (x)$ is open.

The above proof also pointed out that for every ${\lambda}_{0}\in \u2102$, ${R}_{x}$ is defined in the open disk of radius $\frac{1}{\parallel {R}_{x}({\lambda}_{0})\parallel}$ centered in ${\lambda}_{0}$.

We now prove the analyticity of the .

Taking inverses^{} on the equality (3) above one obtains

$${R}_{x}(\lambda )={(e-(\lambda -{\lambda}_{0}){R}_{x}({\lambda}_{0}))}^{-1}\cdot {R}_{x}({\lambda}_{0})$$ |

Again, by the Neumann series (http://planetmath.org/NeumannSeriesInBanachAlgebras), one obtains

$${R}_{x}(\lambda )=\left[\sum _{n=0}^{\mathrm{\infty}}{R}_{x}{({\lambda}_{0})}^{n}{(\lambda -{\lambda}_{0})}^{n}\right]\cdot {R}_{x}({\lambda}_{0})=\sum _{n=0}^{\mathrm{\infty}}{R}_{x}{({\lambda}_{0})}^{n+1}{(\lambda -{\lambda}_{0})}^{n}\mathit{\hspace{1em}}\mathrm{\square}$$ |

Title | resolvent function is analytic |
---|---|

Canonical name | ResolventFunctionIsAnalytic |

Date of creation | 2013-03-22 17:29:36 |

Last modified on | 2013-03-22 17:29:36 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 8 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46H05 |

Classification | msc 47A10 |