resolvent matrix

The resolvent matrix of a matrix $A$ is defined as

$R_{A}(s)=(sI-A)^{-1}.$

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 $\mathbb{C}$ . The resolvent $R_{x}$ of an element $x\in A$ is a function from $\mathbb{C}-Sp(x)$ to $A$ given by

$R_{x}(s)=(s\cdot 1-x)^{-1}$

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

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