Category:Definitions/Spectra (Spectral Theory)

This category contains definitions related to Spectra (Spectral Theory).
Related results can be found in Category:Spectra (Spectral Theory).

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 and denote the spectrum of $x$ 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 $A_+$.