Banach Algebra - Spectral Theory

Spectral Theory

Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum of an element x ∈ A, denoted by, consists of all those complex scalars λ such that x − λ1 is not invertible in A. The spectrum of any element x is a closed subset of the closed disc in C with radius ||x|| and center 0, and thus is compact. Moreover, the spectrum of an element x is non-empty and satisfies the spectral radius formula:

Given x ∈ A, the holomorphic functional calculus allows to define ƒ(x) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

When the Banach algebra A is the algebra L(X) of bounded linear operators on a complex Banach space X  (e.g., the algebra of square matrices), the notion of the spectrum in A coincides with the usual one in the operator theory. For ƒ ∈ C(X) (with a compact Hausdorff space X), one sees that:

The norm of a normal element x of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.

Let A  be a complex unital Banach algebra in which every non-zero element x is invertible (a division algebra). For every a ∈ A, there is λ ∈ C such that a − λ1 is not invertible (because the spectrum of a is not empty) hence a = λ1 : this algebra A is naturally isomorphic to C (the complex case of the Gelfand-Mazur theorem).