Definition:Spectrum (Spectral Theory)
Definition
Bounded Linear Operator
Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\C$.
Let $A : X \to X$ be a bounded linear operator.
Let $\map \rho A$ be the resolvent set of $A$.
Let:
- $\map \sigma A = \C \setminus \map \rho A$
We say that $\map \sigma A$ is the spectrum 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.
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:
- $\map \sigma T = \C \setminus \map \rho T$
Unital Algebra
Let $A$ be a unital algebra over $\C$.
Let $x \in A$.
Let $\map {\rho_A} x$ be the resolvent set of $x$ in $A$.
We define the spectrum of $x$, $\map {\sigma_A} x$, by:
- $\map {\sigma_A} x = \C \setminus \map {\rho_A} x$
Non-Unital Algebra
Let $A$ be an algebra over $\C$ that is not unital.
Let $x \in A$.
Let $A_+$ be the unitization of $A$.
We define the spectrum of $x$, $\map {\sigma_A} x$, by:
- $\map {\sigma_A} x = \map {\sigma_{A_+} } {\tuple {x, 0} }$
where $\map {\sigma_{A_+} } {\tuple {x, 0} }$ is the spectrum of $\tuple {x, 0}$ in $A_+$.