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At the moment I am trying to understand "Lectures on Floer homology" By D. Salamon, see

In step 1 of the proof of Lemma 2.4 (page 17) he considers the following operator:

$A=J_0\partial_t+S:W^{1,2}(S^1,R^{2n})\rightarrow L^2(S^1,R^{2n}),$

where $J_0$ is the standard complex structure on $R^{2n}$ and $S:[0,1]\rightarrow \text{Sym}(2n)$ is a smooth path of symmetric matrics chosen in such a way that $A$ is an isomorphism. Now he claims that $L^2(S^1,R^{2n})$ can be decomposed into positive and negative eigenspaces of $A$. If A was a bounded operator, this would be clear to me (using the spectral projection via operator calculus). Unfortunately, $A$ is an unbounded operator and the integrals involved might be ill-defined. Question 1: How does this work?

He then claims that the operators $-A^+$ and $A^-$ generate (quasi-contractive) strongly continuous semi-groups of operators on $E^+$ and $E^-$, where $A^{\pm}$ denote the restrictions of $A$ onto the positive and negative eigenspaces $E^+$ and $E^-$. I am aware of the generation theorem for strongly continuous semi-groups. But I don't actually see how to get the condition $\|(\lambda+\delta)(\lambda-A^-)^{-1}\|\leq 1$ for all $\lambda>-\delta,$ where $-\delta<0$ is an upper bound for the negative spectrum of $A.$ (And correspondingly for $A^+.$) (As a reference I use Engel/Nagel: One-Parameter Semigroups for Linear Evolution Equations, Chapter II, Corollary 3.6.) There is also a version of the generation theorem for symmetric operators (A is symmetric w.r.t. the $L^2$ inner product), provided one can find a bound $\langle x,A^{-}x\rangle_{L^2([0,1],R^{2n})}\leq -\delta$ for $x\in W^{1,2}([0,1],R^{2n})\cap E^-.$ Question 2: What are the correct arguments?

Could anyone please help me with these issues and/or provide appropriate references for the two claimed statements?

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1 Answer 1

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Well, at first sight, the operator is claimed to be self-adjoint and injective. This means that it is unitary equivalent to a multiplication operator by the spectral theorem, and you can split the space using the spectral projections corresponding to the intervals $(-\infty,0)$ and $(0,\infty)$. Weidmann: Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics, 68. Springer-Verlag, New York-Berlin, 1980, is a good reference on unbounded selfadjoint oerators.

Then you get what you want: splitting the space, and in one a s.a. operator with negative spectrum, and an other with s.a operator with positive spectrum. The first is dissipative, and minus the second will be also dissipative.

Unfortunately, these arguments are not easy if you see them for the first time. But people use them quite often...

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Thank you! I checked the reference and I think I understood the stuff. – OrbiculaR Oct 25 '11 at 18:42

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