I'm wondering whether there is such a thing as a "residue theorem for holomorphic operator-valued functions". More precisely, I want to evaluate an integral of the form
$P:=\int_{\Gamma} (A(\lambda) - \lambda)^{-1} d \lambda$
where each $A(\lambda)$ is a closed operator and $\Gamma$ encloses an eigenvalue $\lambda_0$ of the holomorphic operator pencil $A(\lambda) - \lambda$, i.e., for some eigenfunction $u_0$ we have $A(\lambda_0)u_0 - \lambda_0u_0=0$.
In the case of a $\lambda$-independent $A$ the operator $P$ is (up to a constant) the well-known Riesz Projection corresponding to $\lambda_0$. If (and how) this can be generalized to a $\lambda$-nonlinear eigenvalue problem is precisely what my question is concerned with.
Thanks for any help in advance!