Definition:Resolvent Set/Densely-Defined Linear Operator

Definition

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.

Note that if $\map D T = \HH$ and $T$ is bounded, we realise we may have a conflict with the definition of its resolvent set as a bounded linear operator.

In Resolvent Set of Bounded Linear Operator equal to Resolvent Set as Densely-Defined Linear Operator, it is shown that no such conflict occurs and the two notions of spectrum coincide.

Also see

• Definition:Resolvent Set of Bounded Linear Operator

• Results about resolvent sets of densely-defined linear operators can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.3$: The Spectrum of Closed Unbounded Self-Adjoint Operators