Definition:Resolvent Set/Unital Algebra

Definition

Let $A$ be a unital algebra over $\C$.

Let $x \in A$.

Let $\map G A$ be the group of units of $A$.

Let:

$\map {\rho_A} x = \set {\lambda \in \C : \lambda {\mathbf 1}_A - x \in \map G A}$

We call $\map {\rho_A} x$ the resolvent set of $x$ in $A$.

Also see

• Results about resolvent sets in unital algebras can be found here.

Sources

• 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $4.9$: Spectrum relative to a subalgebra