> [!definition] > > Let $\alg$ be a [[Banach Algebra|Banach algebra]] and $x \in \alg$. The **resolvent** (set) of $x$ is the complement of its [[Spectrum|spectrum]]. If $\lambda \not\in \spec{x}$, then the **resolvent** (operator) of $x$ is the operator > $ > R(\lambda) = R_x(\lambda) = (\lambda e - x)^{-1} > $ > which is analytic on the resolvents[^1]. [^1]: See [[Power Series|power series]].