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 $A_+$.

Sources

• 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $4.5$: The spectrum