# A question on strongly continuous semi-groups

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

http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf

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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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.