I am reading [Link](https://doi.org/10.1007/BF01360915 "Tosio Kato: Wave operators and similarity for some non-selfadjoint operators"). 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?



How to go from a potential resolvent to the associated operator
Such operator families $R(\zeta)$ are called pseudoresolvents. The result you are looking for is, for example, proved in Chapter III, Proposition 4.6 of the book "One-parameter semigroups for linear evolution equations" by K. J. Engel and R. Nagel:


Google books link: <http://books.google.de/books?id=xcYVVSyAOkgC&pg=PA207&lpg=PA207&dq=pseudoresolvent+semigroup&source=bl&ots=qkEyL0hmVG&sig=YOIV5d9z4PzTHUgUMzds8vFKVL4&hl=de&ei=MBxMTq0Cj8myBqb1yMoB&sa=X&oi=book_result&ct=result&resnum=10&ved=0CGIQ6AEwCQ#v=onepage&q=pseudoresolvent%20semigroup&f=false>


