Spectrum (functional Analysis) - Spectrum of A Unital Banach Algebra

Spectrum of A Unital Banach Algebra

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ(x) (or more explicitly σ_B(x)) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B. This extends the definition for bounded linear operators B(X) on a Banach space X, since B(X) is a Banach algebra.