My ultimate aim is to "hopefully" show that under certain assumptions, the psp of an element in a Banach algebra consists entirely of Riezs points. This is just one step closer to that goal. I have seen it done by Schaefer for bounded linear operators, but I'm hoping to transport that result into Banach algebras.

Thanks for the comments. I realise I was not very clear in the theorem. I am actually assuming that psp(a) contains isolated points only and that $p(λ_i,a)= \frac{1}{2\pi i}\int_{\gamma_i}(a-\alpha)^{-1} d\alpha$ with $\gamma_i$ a circle around $\lambda_i$ which does not contain any other points of $\sigma(a)$.