I am learning a new topic about holomorphic functional calculus, however I cannot understand this definition.
Definition: Let $T$ a bounded operator in a Hilbert Space, and $f$ a holomorphic fonction in a open set U, we set \begin{equation} f(T)=\frac{1}{2i\pi}\int_{\partial U}(z-T)^{-1}f(z)dz \end{equation} What I don't understand, it's in deed the definition of the integral, How I can integrate $(z-T)^{-1}$?, or I am only evaluating $f(z)$ in the resolven of $T$? or it is a multiplication?
Maybe my question is very basic, but I don't find any book that explains in a better way.
It is better to write
$$\begin{equation} f(T)=\frac{1}{2i\pi}\int_{\partial U}f(z)(z-T)^{-1}dz \end{equation}$$
instead of
$$\begin{equation} f(T)=\frac{1}{2i\pi}\int_{\partial U}(z-T)^{-1}f(z)dz \end{equation}.$$
Let $g(z):=f(z)(z-T)^{-1}.$ Then $g$ is continuous on $ \partial U$. Observe that we have $\partial U \subset \rho(T)$.
Hence
$$f(T)= \frac{1}{2i\pi}\int_{\partial U}g(z)dz.$$