I am reading http://www.springerlink.com/content/l76542r216362714/. The author appears to use the following fact:

Let $H$ be a Hilbert space. For every $\zeta \in \mathbb{C}\setminus\mathbb{R}$ we have a bounded operator $R(\zeta): H \to H$. We also know that

(1) $R(\zeta)$ has nullity $0$

(2) $R(\zeta)$ has dense range

(3) $R(\zeta)$ satisfies the first resolvent identity $R(\zeta) - R(\zeta') = (\zeta-\zeta')R(\zeta)R(\zeta')$.

Then we claim that there exists a densely defined operator $T: H \to H$ such that $\sigma(T) \subset \mathbb{R}$ and for $\zeta \in \mathbb{C}\setminus\mathbb{R}$ the resolvent at $\zeta$ is $R(\zeta)$.

How is this proved? Alternatively, does anyone know a reference where this is proved?