Definition:Spectrum (Spectral Theory)/Non-Unital Algebra

Definition
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 $A_+$.