Abstract
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces whose intersection contains a fixed vector space \(\mathscr {D}\). In the case when \(\mathscr {D}\) is dense in one of the Hilbert spaces (but not necessarily in the other), we make precise an operator-theoretic linking between the two Hilbert spaces. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and the operator theory of reflection positivity.
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Notes
As a map between sets, J is the inclusion map \(C\left( X\right) \hookrightarrow L^{2}\left( X,\mu \right) \). However, we are considering \(C\left( X\right) \subset L^{2}\left( X,\lambda \right) \) here, and so J is not an inclusion map between Hilbert spaces because the inner products are different. Perhaps “pseudoinclusion” would be a better term.
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Acknowledgements
The co-authors thank the following colleagues for helpful and enlightening discussions: Professors Daniel Alpay, Sergii Bezuglyi, Ilwoo Cho, Paul Muhly, Myung-Sin Song, Wayne Polyzou, and members in the Math Physics seminar at the University of Iowa and the Operator Theory seminar at Cal Poly.
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Jorgensen, P., Pearse, E. & Tian, F. Unbounded operators in Hilbert space, duality rules, characteristic projections, and their applications. Anal.Math.Phys. 8, 351–382 (2018). https://doi.org/10.1007/s13324-017-0173-9
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DOI: https://doi.org/10.1007/s13324-017-0173-9