I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph "General Eigenfunction Expansions and Unitary Representations of Topological Groups", second chapter or alternatively his paper "Allgemeine Eigenfunktionsentwicklungen. Spektraldarstellung abstrakter Kerne. Eine Verallgemeinerung der Distributionen auf Lie'schen Gruppen" (Bullentin de L'Academie Polonaise des Sciences.
Serie des sci. math.
astr. et phys. 7 (8 1959), pp. 471-479.
) which contains basically the same proof.
Let $\Phi\subset H\subset \Phi'$ be a rigged Hilbert space, $\Phi$ nuclear. We want to prove, that there exists a complete set of generalized eigenfunctions in $\Phi'$.
The basic idea is the following: We use the Integral formulation of the Neumann spectral theorem, i.e. every normal operator on a separable Hilbert space $H$ is unitary equivalent to a multiplication operator on a direct integral of Hilbert Spaces (which is basically an $L^2$-type complex-valued function space), the latter shall be called $\hat{H}$ and the unitary transformation $F:\varphi\rightarrow \varphi(x)$. The key element in the proof is now, that we can prove, that evaluating some $F\varphi$ at a point of the spectrum, say $\lambda$, is a continuous mapping into the complex numbers and therefore associated with some element of the dual space. Concretely, Maurin "shows" that that the mapping $F(\lambda):\varphi\rightarrow\varphi(\lambda)$ is continuous for every $\lambda$ up to a set of measure zero.
I highly dout that this mapping is even well-defined, as the functions in the direct integral of Hilbert spaces are only defined up to an equivalence class, i.e. they can differ on a set of measure zero.
Am I overlooking something trivial? Is there a simple way to patch this up?
Is there some reference which solves this issue? I have heard that the proof given by Gelfand and Vilekin also has a similar issue.
I would greatly appreciate any help, I am very stuck at the moment.
Best Answer
In this discussion, there is a pointer to a paper by G. G. Gould (J. LMS, 1968) which supposedly gives a correct but hard to read proof.
This is the paper: \bib{MR0230148}{article}{ author={Gould, G. G.}, title={The spectral representation of normal operators on a rigged Hilbert space}, journal={J. London Math. Soc.}, volume={43}, date={1968}, pages={745--754}, issn={0024-6107}, review={\MR{0230148 (37 #5713)}}, }