Definition:Spectrum (Spectral Theory)/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.

Let $\map \rho T$ be the resolvent set of $\struct {\map D T, T}$.

We define the the spectrum of $T$, $\map \sigma T$, by:

$\map \sigma T = \C \setminus \map \rho T$

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

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

Also see

• Definition:Spectrum of Bounded Linear Operator

• Results about spectra 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