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I am reading 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?

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1 Answer 1

up vote 3 down vote accepted

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:

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Thanks, I'm not sure why it didn't occur to me to set $A=R(i)^{−1}+i$... – Yakov Shlapentokh-Rothman Aug 17 '11 at 20:13

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