Here is a little theorem that I'm trying to prove. I haven't seen it in literature before, but I think the applications will be quite useful, particularly in the context of Banach algebras.

Denote by psp($a$) the peripheral spectrum of $a\in A$.

$\underline{\textrm{Theorem}}$

Let $A$ be a Banach algebra with $a\in A$, $r(A)>0$ and psp($a$)={$\lambda_1,\lambda_2,...,\lambda_k$}.

Then $a=a_0+∑_1^k λ_i p_i $

where $a_0=a(1-p)$ and $p$ is the spectral projection relative to $a$ and {$λ_1,λ_2,⋯,λ_k$}.

The following is as far as I got, and any suggestions would be much appreciated.

$\underline{\textrm{Proof}}$:

It is clear that $a=a(1-p)+ap=a_0+ap$.

By Cauchy's Theorem for Multiply Connected Domains, we have that $p=p(λ_1,a)+p(λ_2,a)+...+p(λ_k,a)=p_1+p_2+...+p_k$

(if we denote $p(λ_i,a)$ by $p_i$).

Thus, $a=a_0+\sum_1^k ap_i$.

What's left to prove is that $ap_i=aλ_i$ for $i=1,...,k$.

I have thought about using the notion of simple poles, which works very nicely if we consider $a$ to be quasi inessential (analogous to quasi compact). But we don't have this in our case.