Precise assumption in spectral theorem of unbounded operators

functional-analysishilbert-spacesoperator-theoryspectral-theoryunbounded-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?

Best Answer

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

  2. Yes.

  3. This is done with a lot of detail in section 5.6 in Kadison-Ringrose Fundamental of the Theory of Operator Algebras, particularly Theorems 5.6.12 and 5.6.26. There's also a significant amount of detail in section X.4 in Conway's A Course in Functional Analysis.

Related Question