Definition:Spectrum (Spectral Theory)/Unital Algebra

Definition

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:

$\map {\sigma_A} x = \C \setminus \map {\rho_A} x$

Also see

• Results about spectra in unital algebras can be found here.

Sources

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