right definition of correct space of domain and range for a self-adjoint Operator at first, I'm quite sure it's not necessary to pay too much attention to the way the Operator is defined, it's rather important which spaces to choose to obtain a self-adjoint operator 
I've got a problem with underständing the definition of the spaces of domain and range. Maybe someone is familiar with problems like that or even know it. In addition I should note that the following is part of a book (Joel Smoller-Shock Waves and Reaction-Diffusion Equations 106/107)
Okay, here it begins:
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Given is the operator $Pu = Au + au= \sum_{i,j=1}^N (a_{ij}(x)u_{x_{i}})_{x_{j}}+a(x)u$
  defined in a bounded domain $\Omega \subset \mathbb{R}^n$, $\partial\Omega$ smooth.
  We assume the coefficients of P are smooth and $a_{ij}=a_{ji}$ in $\Omega$
  Furthermore we assume A is strong elliptic in \Omega,in the sense that there is an $\alpha >0$ such that 
  $\sum_{i,j=1}^Na_{ij}(x)\xi_i \xi_j>=\alpha |\xi|^2 , x \in \Omega$ 
We consider the operator P together with homogeneous boundary conditions of the form
  $\alpha(x)\frac{du}{dn}+\beta(x)u=0$ on $\partial\Omega$ where $\alpha(x)\geq 0 , \beta(x)\geq 0 , \alpha(x)^2 + \beta(x)^2 \ne 0$
  We regard $P$ as an Operator acting on functions in $W^2(\Omega)$ which satisfy the boundary conditions on $\partial\Omega$, into $L_2(\Omega)$ 

The statement now is that P is a self-adjoint operator!
But this is my problem, to check the self-adjointness. Formally I need a hilbert-space $H$ and an operator $L:D(L)\subset H \rightarrow H$ defined on a dense subset $D(L)\subset H$
But in this case, I've got two different hilbertspaces where D(P) is (probably) a dense subset of $W^2(\Omega)$  but dense in $L^2(\Omega)$.
I hope someone could help me with that.
 A: Showing an operator such as $P$ is symmetric on its domain $\mathcal{D}(P)$ is the first place to start. All selfadjoint operators are symmetric, but not necessarily the other way around. So, first show that
$$
                 (Pf,g)=(f,Pg),\;\;\; f,g \in \mathcal{D}(P).
$$
The inner-product should be the $L^{2}(\Omega)$ inner-product. The domain $\mathcal{D}(P)$ consists of all $u\in W^{2}(\Omega)$ which satisfy the boundary condition $\alpha \frac{\partial u}{\partial n}+\beta u=0$. And it is useful to show $P$ is semibounded if it is; that is, is there a constant $m$ such that
$$
               (Pf,f) \ge m\|f\|^{2},\;\;\; f \in \mathcal{D}(P)?
$$
It's not important to establish that $\mathcal{D}(P)$ is dense in $W^{2}(\Omega)$--in fact, that won't be true. But it is important to show that $\mathcal{D}(P)$ is dense in $L^{2}(\Omega)$. If $P$ is not densely-defined in $L^{2}(\Omega)$, then an adjoint $P^{\star}$ isn't well-defined, and selfadjoint has no meaning. The underlying Hilbert space $H$ is $L^{2}(\Omega)$. The fact that $W^{2}(\Omega)$ is a Hilbert space is not directly relevant; you're just trying to find a domain $\mathcal{D}(P)$ on which $P$ will be a densely-defined selfadjoint linear operator. So, the setting is
$$
                P : \mathcal{D}(P)\subset L^{2}(\Omega)\rightarrow L^{2}(\Omega),
$$
and you want to choose that domain so that $P=P^{\star}$, where the adjoint is taken in $L^{2}(\Omega)$.
It may seem hard to prove that the domain is dense. However, if you can show that $P$ is symmetric and that $P-\lambda I$, $P-\overline{\lambda}I$ are surjective for some $\lambda \in \mathbb{C}\setminus\mathbb{R}$, then the domain must be dense and $P$ must be selfadjoint. If $P$ is bounded below by some real $m$ as above, one need only verify that $P+\lambda I$ is surjection for some real $\lambda > -m$. Either verification usually comes down to classical solvability for the PDE.
