# Precise assumption in spectral theorem of unbounded operators

The most general version of the spectral theorem I am aware of is the spectral theorem for unbounded normal operators (firstly proven by von Neumann in 1932, I think). An operator $$T:\mathcal{D}(T)\to\mathcal{H}$$ in some Hilbert space $$\mathcal{H}$$ is called normal, if

$$TT^{\ast}=T^{\ast}T.$$

Note that this is an equlity on the level of operators, which means that we require that

$$\mathrm{D}(TT^{\ast})=\{\psi\in\mathcal{D}(T^{\ast})\mid T\psi\in\mathcal{D}(T)\}\stackrel{!}{=}\{\psi\in\mathcal{D}(T)\mid T\psi\in\mathcal{D}(T^{\ast})\}=\mathcal{D}(T^{\ast}T)$$

as well as

$$TT^{\ast}\psi=T^{\ast}T\psi,\hspace{2cm}\forall \psi\in\mathcal{D}(TT^{\ast}).$$

Roughly speaking, the spectral theorem for general normal (possibly unbounded) operators states in its measure-theoretic formulation the following:

Let $$T$$ be a normal operator $$T:\mathcal{D}(T)\to\mathcal{H}$$. Then there exists a unique spectral measure $$P:\mathcal{B}(\sigma(T))\to\mathcal{B}(\mathcal{H})$$, where $$\mathcal{B}(\sigma(T))$$ denotes the Borel $$\sigma$$-algebra on the spectrum $$\sigma(T)$$ and $$\mathcal{B}(\mathcal{H})$$ the set of bounded operators on $$\mathcal{H}$$, such that $$\mathcal{D}(T)=\bigg\{\psi\in\mathcal{H}\,\bigg\vert\,\int_{\sigma(T)}\,\vert\lambda\vert^{2}\,\mathrm{d}\langle\psi,P_{\lambda}\psi\rangle\bigg\}$$ and $$T=\int_{\sigma(T)}\,\lambda\,\mathrm{d}P_{\lambda}.$$

Now, unfortunately, it is quite hard to find a discussion of the spectral theorem in this general version in the literature and hence, I am unsure about the precise requirements: In particular, I have the following short questions:

1. Does one have to assume separability of the Hilbert space $$\mathcal{H}$$?
2. Shouldn't one assume more precisely that $$T$$ is closed and densley-defined? As far as I know, there is a theorem stating that if $$T$$ is densely-defined and closed, then $$TT^{\ast}$$ is itself densely-defined and self-adjoint and I guess that this is what we need in order to proof the spectral theorem.
3. Is there any other requirement for the theorem to hold? Does one have a good literature, which treats the spectral theorem in this level of generality?

1. No. You can always write $$H$$ as a direct sum of pairwise orthogonal separable reducing subspaces for $$T$$.