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

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

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

If $A$ is commutative and $s\notin Sp(x)\cup Sp(y)$, then $R_x(s)-R_y(s)=R_x(s)R_y(s)(x-y)$.