# Error in Maurins proof for the nuclear spectral theorem?

I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph [2], second chapter or alternatively his paper [1] 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.

References

[1] Krzysztof Maurin,"Allgemeine Eigenfunktionsentwicklungen. Spektraldarstellung abstrakter Kerne. Eine Verallgemeinerung der Distributionen auf Lieschen Gruppen". (German) Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 7, 471-479 (1959), MR114133, Zbl 0106.08903.

[2] Krzysztof Maurin, General eigenfunction expansions and unitary representations of topological groups. (English) Monografie Matematyczne. 48. Warszawa: PWN - Polish Scientific Publishers. 367 p. (1968), MR247377, Zbl 0185.39001.

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:
Gerald G. Gould, "The spectral representation of normal operators on a rigged Hilbert space". (English) Journal of the London Mathematical Society 43, 745-754 (1968), MR0230148, Zbl 0159.43401.

• Thank you, this paper seems to resolve the question. The proof seems to be quite technical, but thats fine I guess. I am not sure whether I maybe have found a simpler solution, but I will proofread it first. Sep 4, 2014 at 22:10
• @Daniel If you found a simpler solution, be sure to keep us posted :) Sep 4, 2014 at 23:46
• @Daniel did you find a simpler solution? Jan 12, 2018 at 0:53
• Any update @Daniel Sep 30, 2020 at 13:07

Perhaps Vol 3 by Gel'fand and Shilov could be helpful here instead of Vol 4?

I. M. Gel’fand, G. E. Shilov, Generalized functions. Vol. 3. Theory of differential equations. Translated from the Russian by Meinhard E. Mayer. Reprint of the 1967 original published by Academic Press. (English) Providence, RI: AMS Chelsea Publishing (ISBN 978-1-4704-2661-3/hbk). pp. x+222 (2016), MR3468845, Zbl 1339.01008.

Gel'fand and Shilov take a slightly different approach there, deriving the generalized eigenfunctions as, essentially, a distributional derivative (in $$\Phi'$$) of the spectral projections $$E_\lambda f$$ of an arbitrary function $$f\in H$$. They then show completeness in the subspace $$\{E_\lambda f\}_{\lambda}$$ using some theory on the a.e. recovery of a function of bounded variation from its Radon-Nikodym derivative. The general case is handled by expressing $$H$$ as a direct sum of such subspaces.

There are many related constructions, because of the connection to wave-packets (in scattering) and the resolvent operator. See, e.g., the following paper

Nicolas A. Derzko, "Generalized eigenfunctions and real-axis limits of the resolvent", Transactions of the American Mathematical Society 174 (1972), 489-506 (1973), MR0310684, Zbl 0221.47015, (available also here).