Definition:Resolvent Set/Bounded Linear Operator

Definition
Let $\struct {X, \norm \cdot}$ be a Banach space over $\C$.

Let $A : X \to X$ be a bounded linear operator.

Let $I : X \to X$ be the identity mapping on $X$.

Let $\map \rho A$ be the set of $\lambda \in \C$ such that $A - \lambda I$ is invertible in the sense of a bounded linear transformation

We call $\map \rho A$ the resolvent set of $A$.

Also see

 * Resolvent Set of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator