# resolvent matrix

Note: $I$ is the identity matrix^{} and $s$ is a complex variable. Also note that ${R}_{A}(s)$ is undefined on $Sp(A)$ (the spectrum of $A$).

More generally, let $A$ be a unital algebra over the field of complex numbers^{} $\u2102$. The *resolvent* ${R}_{x}$ of an element $x\in A$ is a function from $\u2102-Sp(x)$ to $A$ given by

$${R}_{x}(s)={(s\cdot 1-x)}^{-1}$$ |

where $Sp(x)$ is the spectrum of $x$: $Sp(x)=\{t\in \u2102\mid t\cdot 1-x\text{is not invertible in}A\}$.

If $A$ is commutative^{} and $s\notin Sp(x)\cup Sp(y)$, then ${R}_{x}(s)-{R}_{y}(s)={R}_{x}(s){R}_{y}(s)(x-y)$.

Title | resolvent matrix |
---|---|

Canonical name | ResolventMatrix |

Date of creation | 2013-03-22 13:36:20 |

Last modified on | 2013-03-22 13:36:20 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 8 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 47A10 |

Classification | msc 15A15 |

Defines | resolvent |