Let $\Gamma$ be a differentiable Jordan curve in the resolvent set $\rho(A)$ of the self-adjoint operator $A$. How does one show $\chi_\Omega(A) = \int_\Gamma R_A(z) dz$, where $\Omega = Int \Gamma \cap \mathbb R$ and $R_A(z)$ is the resolvent of $A$ at $z$.
Of course, this looks like Cauchy's integral formula (isnt there a $1/2\pi i$-factor missing?). The RHS is a Banach-space valued integral (the operators $R_A(z)$ are bounded operators) and somewhere i've read that function theory easily extends to the setting of general Banach algebras, but i dont feel confident enough to argue why the above formula holds. How could one use the functional calculus from the spectral theorem (for self-adjoint operators) here?