I have been reading Martin Schechter's Principles of Functional Analysis and was wondering if anyone was aware of a more rigorous proof of the following theorem:
$\mathbf{Theorem\ 9.4:}$ if $\lambda$, $\mu$ are in $\rho(a)$, then $$(\lambda e - a)^{-1} -(\mu e-a)^{-1}=(\mu -\lambda)(\lambda e - a)^{-1}(\mu e-a)^{-1}$$
If $$|\lambda-\mu|\cdot\|(\mu e-a)^{-1}\|\lt1,$$ then $$(\lambda e - a)^{-1}=\sum_1^\infty(\mu -\lambda)^{n-1}(\mu e-a)^{-n}$$
Here, $\rho(a)$ denotes the resolvent of an element $a$ in a Banach algebra. I have The first equation figured out but the second part I am unsure of how to prove. Thanks in advance!
The series is convergent because of the assumption that $|\lambda -\mu| \|(\mu e-a)^{-1}| <1$. Denote the sum by $S$ and just calculate $(\lambda e -a)S$ and $S(\lambda e -a)$. You will get $e$ for both.