How to go from a potential resolvent to the associated operator - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T15:41:15Z http://mathoverflow.net/feeds/question/73092 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73092/how-to-go-from-a-potential-resolvent-to-the-associated-operator How to go from a potential resolvent to the associated operator Yakov Shlapentokh-Rothman 2011-08-17T19:07:51Z 2011-08-17T20:00:52Z <p>I am reading <a href="http://www.springerlink.com/content/l76542r216362714/" rel="nofollow">http://www.springerlink.com/content/l76542r216362714/</a>. The author appears to use the following fact:</p> <p>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</p> <p>(1) $R(\zeta)$ has nullity $0$</p> <p>(2) $R(\zeta)$ has dense range</p> <p>(3) $R(\zeta)$ satisfies the first resolvent identity $R(\zeta) - R(\zeta') = (\zeta-\zeta')R(\zeta)R(\zeta')$.</p> <p>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 $R(\zeta)$. </p> <p>How is this proved? Alternatively, does anyone know a reference where this is proved? </p> http://mathoverflow.net/questions/73092/how-to-go-from-a-potential-resolvent-to-the-associated-operator/73097#73097 Answer by Heiko for How to go from a potential resolvent to the associated operator Heiko 2011-08-17T20:00:52Z 2011-08-17T20:00:52Z <p>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:</p> <p>Google books link: <a href="http://books.google.de/books?id=xcYVVSyAOkgC&amp;pg=PA207&amp;lpg=PA207&amp;dq=pseudoresolvent+semigroup&amp;source=bl&amp;ots=qkEyL0hmVG&amp;sig=YOIV5d9z4PzTHUgUMzds8vFKVL4&amp;hl=de&amp;ei=MBxMTq0Cj8myBqb1yMoB&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=10&amp;ved=0CGIQ6AEwCQ#v=onepage&amp;q=pseudoresolvent%20semigroup&amp;f=false" rel="nofollow">http://books.google.de/books?id=xcYVVSyAOkgC&amp;pg=PA207&amp;lpg=PA207&amp;dq=pseudoresolvent+semigroup&amp;source=bl&amp;ots=qkEyL0hmVG&amp;sig=YOIV5d9z4PzTHUgUMzds8vFKVL4&amp;hl=de&amp;ei=MBxMTq0Cj8myBqb1yMoB&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=10&amp;ved=0CGIQ6AEwCQ#v=onepage&amp;q=pseudoresolvent%20semigroup&amp;f=false</a></p>