Let $H$ be a Hilbert space and $T:H\rightarrow H$ is a bounded linear operator. By holomorphic functional calculus, we can write $\displaystyle T=\oint_{C}\frac{z}{z-T}dz$, where $C$ is the contour enclosed the spectrum of $T$. Now choose $a\in\mathbb{C}$ so that it is outside the contour, $\lVert T-aI \rVert<|z-a|$, and $T-a$ is bijective. Then we write:
$T\displaystyle =\oint_{C}\frac{z}{z-a-T+a}dz=\oint_{C}\frac{z}{z-a}\frac{1}{1-(\frac{T-a}{z-a})}dz$,
by Neumann series:
$\displaystyle T=\oint_{C}\frac{z}{z-a}\sum_{n=0}^{\infty}(z-a)^{-n}(T-a)^{n}dz$.
Now let $v\in H$, the first question is, can we do the following?
$\displaystyle Tv=\oint_{C}\frac{z}{z-a}\sum_{n=0}^{\infty}(z-a)^{-n}(T-a)^{n}v \,dz$
Assume we can do that, let $(T-a)^{n}v=u_{n}\in H$, then we have:
$\displaystyle Tv=\oint_{C}\frac{z}{z-a}\sum_{n=0}^{\infty}(z-a)^{-n}u_{n}\,dz$.
The second question is, can we do the following?
$\displaystyle Tv=\sum_{n=0}^{\infty}\oint_{C}\frac{z}{(z-a)^{n+1}}\,dz\,u_{n}$
