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 as a bounded linear transformation

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

Also see

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

• Results about resolvent sets of bounded linear operators can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $14.1$: The Resolvent and Spectrum