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I made a first draft of a page about unbounded operators, the battle plan contains some basic definitions, explanation of some subtleties of domain issues and what it means to be affiliated to a von Neumann algebra. Right now, only the rigged Hilbert space page refers to it.
Great to have that topic opened. It is good to have however the central definition clear, here it is a bit lost among many descriptions which are correct of course. My advice. If you already wrote so much you could have written the very definition (something like: an unbounded operator on a Hilbert space H is a linear operator defined on a dense subspace of H; bounded operators are an example). The entry makes an impression that this is a SUBclass of a notion of a linear operator on H, and in fact it is an EXTENSION of that notion by allowing dense subspaces of definition.
It is good to have however the central definition clear…
I did not notice that the whole “definition” paragraph was missing, it’s there now.
If you already wrote so much…
It’s only a fraction of what I would like to write, but the subject is clearly too vast for one page…my motivation is twofold:
Demonstrate with a few examples that a purly formal manipulation of unbounded operators can go completly wrong like in Nelson’s counterexample: If A and B can be restricted to a common domain of essential selfadjointness and commute on that, then the elements of the generated semigroups need not commute. Completly violates the “intuition” build from quantum mechanics.
Explain the spectral theorem via affiliation with the abelian von Neumann algebra that is generated by the spectral projections.
That will already take a few hours to do.
Great! Go on, I would help, have I not being in such a time/troubled situation with lots of deadlines, demands and all of that within a rare opportunity to do some new work done in the environment of IHES which I am unfortunately leaving in only 10 days (provided the Island's volcano-mediated air-traffic disturbance permits).
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