# Definition:Resolvent Set

## Definition

### Bounded Linear Operator

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 $A$.

### Densely-Defined Linear Operator

Let $\HH$ be a Hilbert space over $\C$.

Let $\struct {\map D T, T}$ be a densely-defined linear operator.

We define the resolvent set of $T$, $\map \rho T$, as the set of $\lambda \in \C$ for which:

$T - \lambda I$ is injective, $\map {\paren {T - \lambda I} } {\map D T}$ is everywhere dense in $\HH$, and $\paren {T - \lambda I}^{-1}$ is bounded.

### Unital Algebra

Let $A$ be a unital 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 $A$.