I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L^2(G,\mu_H)$ decomposes as \begin{equation} \int^{\oplus}_{\hat{G}} H_\lambda^*\otimes H_\lambda \, d\mu_P. \end{equation} On the other hand, by the nuclear spectral theorem (btw, do you know a nice reference with rigorous proof?), for $A:D(A) \subseteq H \to H$ being a densely defined self-adjoint operator, there exists a dense nuclear subspace $\Phi(A) \subseteq H$, s.t. $H$ decomposes as a direct integral over generalized eigenspaces $\tilde{H}_\lambda$ of $A$, s.t. we get maps $\Phi(A) \to \tilde{H}_\lambda$ for every $\lambda \in \sigma(A)$. Now, I'm looking for such an operator $A$, such that the support of the above Plancherel measure coincides with the generalized spectrum of $A$. I know that the Laplacian is the obvious choice here, but I'm not sure if this is always possible (self-adjointness, etc.). Also, do you know a good reference where harmonic analysis and the theory of generalized functions are combined?
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4$\begingroup$ Reference for harmonic analysis and the theory of generalized functions: Gelfand and Vilenkin, Generalized Functions Vol. 4. To be republished by AMS in May. $\endgroup$– PassingThruCommented Apr 12, 2016 at 18:27
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$\begingroup$ It's getting republished? Excellent news! $\endgroup$– user70229Commented Apr 12, 2016 at 18:56
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6$\begingroup$ Gelfand & Vilenkin's proof has a gap found by the translator (Amiel Feinstein), and this is stated in a footnote. G. G. Gould gave a proof filling the gap, and in the more general case of a normal operator: jlms.oxfordjournals.org/content/s1-43/1/745.full.pdf $\endgroup$– Robert FurberCommented Apr 13, 2016 at 9:49
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3$\begingroup$ I think K. Maurin's General eigenfunction expansions and unitary representations of topological groups (PWN, 1968) might be a helpful reference. $\endgroup$– Igor KhavkineCommented Apr 14, 2016 at 12:18
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