Definition:Resolvent Set/Unital Algebra

Definition
Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.

Let $x \in A$.

Let $\map G A$ be the group of units of $A$.

Let:
 * $\map {\rho_A} x = \set {\lambda \in \C : \lambda {\mathbf 1}_A - x \in \map G A}$

We call $\map {\rho_A} x$ the resolvent set of $x$ in $A$.