Diagonalization in Banach Algebras and Jordan-Banach Algebras: Diagonalization

Let A be a complex Banach or Jordan-Banach algebra. To study the properties of the spectrum function x 7→ Sp x we use the so called Hausdorff distance on compact sets of the complex plane C defined by
12 ½
λ∈σ2
∆(σ , σ ) = max sup {dist(λ, σ )}, sup {dist(λ, σ )}
1
λ∈σ1
2
¾
where dist(λ, σ ) = inf {|λ − µ| : µ ∈ σ } is the distance of the point λ to the compact set σ (see [1, p. 48]). In this paper, we intend to prove that if the spectrum of an element a ∈ A is finite and the function x 7→ Sp x is lipschitzian at a, that is ∆ ¡ Sp(a + x), Sp(a)¢ ≤ M ||x||, then a is diagonalizable; in other words we can write a as a linear combination of pro jections. B. Aupetit proved an analogous spectral characterization for idempotents in [3], that is, elements e ∈ A such that e = e (hence in particular Sp e = {0, 1}), this is in
2
fact contained in the proof of [3, Theorem 1.1]. Recall that a Banach algebra A is said to be semisimple if Rad(A) = {0}, where Rad(A) is the Jacobson radical of A. In what follows ρ(x) stands for the spectral radius of the element x, in our case we can write ρ(x) = max{|λ| : λ ∈ Sp x}. We gather now some well-known results on the spectrum [1, 3].