Let $T$ be an element of a Banach Algebra, and $\sigma(T)$ its spectrum.
I don't understand the above definition, specially when I see the following equalities
$$f(T)S=\oint f(z)(z-T)^{-1}S \ dz$$
and
$$Sf(T)=\oint f(z)S(z-T)^{-1} \ dz$$
If we were thinking of operators $B(X)$, then I would understand $f(T)S$ as $f(T)\circ S=\oint f(z)(z-T)^{-1}(S) \ dz$. However, I would not know how we would deduce the 2nd equality...
Any help would be appreciated.