Definition:Continuous Spectrum of Densely-Defined Linear Operator

Definition

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

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

We define the continuous spectrum $\map {\sigma_s} T$ as the set of $\lambda \in \C$ such that:

$T - \lambda I$ is injective, $\map {\paren {T - \lambda I} } {\map D T}$ is everywhere dense in $\HH$ but $\paren {T - \lambda I}^{-1}$ is not bounded.

Also see

• Results about continuous spectrums 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