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At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which is symmetric and does not have $0$ as an eigenvalue. It is then claimed that the operator generates strongly continuous semigroups on the respective eigenspaces.

In fact I would like to understand these notions, but I don't seem to find good introductory texts. Since the topic seems rather old googling mostly gives me very recent result which are in particular much too specific.

Can anyone give me a reference on symmetric unbounded operators on real Hilbert spaces and their corresponding semigroups?

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Did you try Yosida's book on functional analysis? –  t.b. Oct 21 '11 at 18:12
    
It would help if you gave a precise reference to Salamon's notes (link + page number) then one could check more easily if a certain result fits. –  t.b. Oct 21 '11 at 18:36
    
The reference is math.ethz.ch/~salamon/PREPRINTS/floer.pdf, page 17, step 1. –  OrbiculaR Oct 21 '11 at 19:38
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Engel and Nagel! –  Jonas Teuwen Oct 21 '11 at 19:45
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Pazy is also nice, but is more focused on analytic semigroups. By the way, the Ornstein-Uhlenbeck semigroup is very nice and has cute properties. @t.b.: Ping! –  Jonas Teuwen Oct 21 '11 at 20:54
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4 Answers

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Though Engel and Nagel was mentioned in the comments, the short version

http://www.fa.uni-tuebingen.de/research/publications/2006/a-short-course-on-operator-semigroups/A_Short_Course_on_Operator_Semigroups.pdf

Is a great introduction.

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I actually found Engel and Nagel quite enlightening. –  OrbiculaR Oct 25 '11 at 18:42
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You may find it useful to look at Chapter 6 of "Linear Operators and Their Spectra" by E. Brian Davies, though he works on Banach spaces.

His book is available online.

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You can try "Semigroups of Linear Operators and Applications to Partial Differential Equations" by Pazy.

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For a first introduction I liked the chapters on Operator Theory and Semigroups in An Introduction to Partial Differential Equations by Rogers and Renardy. It keeps things simple, explicit and gets the main ideas across. After reading these introductory chapters other functional analysis texts will be much easier to read.

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