Suppose that $ \Omega $ is some domain, and is the disjoint union of two sub-domains $ \Omega_1 $ and $ \Omega_2 $.
I have a few questions about this.
Firstly, is it true that \begin{equation}\label{key} L^2(\Omega) = L^2 (\Omega_1) \oplus L^2( \Omega_2)? \end{equation}
Secondly, if the above equation is true, is it an orthogonal decomposition?
Thirdly, suppose I have a self-adjoint operator $ H $ on $ L^2(\Omega) $. Denote the restriction of $ H $ to $ L^2 (\Omega_j) $ by $ H_j $. Furthermore, suppose that the spectrum of $ H_1 $ and $ H_2 $ are both simple, and we construct an orthonormal eigenbasis for $ L^2 (\Omega_j) $. My question is: will the union of the spectrum of $ H_1 $ and $ H_2 $ coincide with the spectrum of $ H $, and would the spectrum of $ H $ gaurenteed to also be simple?
I am fairly confident that my first two questions are correct, because $ L^2(\Omega_1) $ is a closed subspace of the Hilbert space $ L^2(\Omega) $, but I'm not completelysure if that is true.
Questions 1 and 2: The decomposition is an orthogonal one.
Question 3: This is not true unless $L^2(\Omega_1)$ and $L^2(\Omega_2)$ is an invariant subspace of $H$. If not here is a counterexample for $\Omega=\{1,2\}$, $\Omega_1=\{1\}$, $\Omega_2=\{2\}$: $$ H=\pmatrix{2&1\\1&2}. $$ The restriction of $H$ to both subspaces has spectrum $\{2\}$, but $H$ has a different spectrum.