I have started learning some Operator Theory. I encountered the following definition. I would like to know why it is that the $f(z)$ in the integrand and the $f(a)$ are both labelled as $f$ where it is clear that they are different mappings. I know that this definition is analogous to the Cauchy integral formula but in that case the functions are actually the same. I don't see why this ambiguous notation is a convention. Please advise what the connection is here.
Riesz-Dunford Functional Analysis:
Let $a \in \mathcal{A}$, where $\mathcal{A}$ is a unital Banach algebra, fix open subset $U \subset \mathbb{C}$ such that $\sigma(a) \subset U$. Take analytic function $f: U \to \mathbb{C}$.
Choose a system $\Gamma \subset U$ of closed contours such that $Ind_{\Gamma}(\lambda) = 1$ for all $\lambda \in \sigma(a)$ and $\{ z \in \mathbb{C}:~\text{Ind}_{\Gamma}(z) \neq 0 \} \subset U$.
Define $$f(a) = \frac{1}{2 \pi i}\int_{\Gamma}f(z)(z-a)^{-1}dz$$
Thanks for any assistance.
This is a "functional calculus". The point is that the function $f$, initially defined on complex numbers, can now be extended to suitable members of the Banach algebra. Moreover, this extension will turn out to have some useful properties.
For a concrete example, consider the Banach algebra $\mathcal L(\mathbb C^n)$ of linear operators on $\mathbb C^n$ (with whatever norm you wish), i.e. $n \times n$ matrices, and the analytic function $f(z) = \sqrt{z}$ defined on $\mathbb C \backslash (-\infty, 0]$. This definition gives you a way to define the square root function on all matrices with no eigenvalues in $(-\infty, 0]$.