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Analysis and Mathematical Physics

, Volume 8, Issue 3, pp 351–382 | Cite as

Unbounded operators in Hilbert space, duality rules, characteristic projections, and their applications

  • Palle Jorgensen
  • Erin Pearse
  • Feng Tian
Article
  • 69 Downloads

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.

Keywords

Quantum mechanics Unbounded operator Closable operator Selfadjoint extensions Spectral theory Reproducing kernel Hilbert space Discrete analysis Graph Laplacians Distribution of point-masses Green’s functions 

Notes

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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe University of IowaIowa CityUSA
  2. 2.Department of MathematicsCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  3. 3.Department of MathematicsHampton UniversityHamptonUSA

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