Spectral theory in non-separable Hilbert Spaces I am wondering about what can be said about the spectral theorem for unbounded, self-adjoint operators in a non-separable Hilbert space. There is a comment in this sense to the question "Does spectral theory assume separability", but for compact normal operators. It says that "the space splits into a direct sum of two spaces, one on which the operator vanishes and a separable one which is spanned by the eigenvectors". Does something similar happen for SA unbounded operators? I would greatly appreciate indications for references on this topic.
 A: As mentioned above, it‘s not entirely clear what you want. Here are two suggestions
1)  the space can be split into a direct sum of separable closed subspaces, each of which is invariant under the given operator.
2)  the space can be split as the direct sum of closed invariant subspaces on each of which the operator is bounded.
There are many ways of stating the spectral theorem and these results allow one to deduce them for the non-separable case from the separable or bounded one.  The non-separability isn‘t really an issue.
A: I agree with user131781, but wanted to add that there is a strong form of the spectral theorem which requires separability. Namely: if $A \in B(H)$ is self-adjoint then there is a probability measure $\mu$ on ${\rm sp}(A)$ and a measurable bundle $\mathcal{H}$ of Hilbert spaces over ${\rm sp}(A)$, such that $H$ can be identified with $L^2({\rm sp}(A), \mathcal{H})$ in a way that takes $A$ to multiplication by $x$.
I like this version because it not only "diagonalizes" $A$, it tells you very explicitly how $A$ sits inside of $B(H)$. This can be useful.
It utterly fails in the nonseparable setting because you can have things like multiplication by $x$ acting on $L^2[0,1]\oplus l^2[0,1]$.
