Timeline for Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
Current License: CC BY-SA 4.0
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| Dec 29, 2019 at 14:05 | comment | added | Robert Furber | I'll also add another oft-unmentioned caveat. People coming from the physics literature often expect that "a complete system of eigenvectors" means that for each spectral value there must exist an eigenvector. This is not the case, as can be seen in the multiplication operator $\frac{1}{1+n}$ on $\ell^2$, which has a complete set of eigenvectors but for which there is no eigenvector, not even in the generalized sense, corresponding to the spectral value $0$. | |
| Dec 29, 2019 at 13:39 | comment | added | Robert Furber | It should be mentioned that Amiel Feinstein, who translated that book into English discovered an error in Gelfand-Vilkenin's proof, involving the treatment of sets of measure zero, which was then corrected by the following paper of G. G. Gould: londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/… | |
| Dec 29, 2019 at 11:37 | comment | added | Tobias Diez | @paulgarrett do you know a good reference that discusses self-adjoint extensions in the framework of Rigged Hilbert spaces? | |
| Dec 29, 2019 at 6:55 | comment | added | Nik Weaver | @paulgarrett: thank you, that's a good example. | |
| Dec 29, 2019 at 2:57 | comment | added | paul garrett | @NikWeaver, Some aspects of self-adjoint extensions of (restrictions of) self-adjoint operators (e.g., Friedrichs extensions in the semi-bounded case) are nicely explained in terms of $H^1\to L^2 \to H^{\-1}$ and such, in my opinion. | |
| Dec 28, 2019 at 23:43 | comment | added | Nik Weaver | Serious question, is there any substantive advantage to working with rigged Hilbert spaces other than being able to rigorously interpret the physicists' delta functions? My feeling has always been that the "right" way to make delta functions rigorous is via spectral theory, but that could just be ignorance on my part. | |
| Dec 28, 2019 at 20:17 | history | edited | Vadim Alekseev | CC BY-SA 4.0 |
minor formulation improvement
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| Dec 28, 2019 at 18:09 | history | answered | Vadim Alekseev | CC BY-SA 4.0 |