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 JorgensenEmail author
  • Erin Pearse
  • Feng Tian


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.


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 



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.


  1. 1.
    Arov, D.Z., Dym, H.: On three Krein extension problems and some generalizations. Integral Equ. Oper. Theory 31(1), 1–91 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alpay, D., Jorgensen, P., Levanony, D.: A class of Gaussian processes with fractional spectral measures. J. Funct. Anal. 261(2), 507–541 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alpay, D., Jorgensen, P., Salomon, G.: On free stochastic processes and their derivatives. Stoch. Process. Appl. 124(10), 3392–3411 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bezuglyi, S., Jorgensen, P.E.T.: Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures. Trends in Harmonic Analysis and Its Applications, Contemp. Math. vol. 650, Am. Math. Soc., Providence, RI, pp. 57–88 (2015)Google Scholar
  5. 5.
    Mischa, C., Roberto C.: An Introduction to Functional Analysis, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc, New York: Translated from the Spanish by A. Torchinsky and A. González Villalobos, North-Holland Texts in Advanced Mathematics (1974)Google Scholar
  6. 6.
    Cho, I., Gillespie, T., Jorgensen, P.E.T.: Asymptotic free probability for arithmetic functions and factorization of Dirichlet series. Anal. Math. Phys. 6(3), 255–295 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colombo, F., Sabadini, I.: The \(F\)-functional calculus for unbounded operators. J. Geom. Phys. 86, 392–407 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks, Carus Mathematical Monographs, vol. 22. Mathematical Association of America, Washington, DC (1984)Google Scholar
  9. 9.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988, Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience PublicationGoogle Scholar
  10. 10.
    Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn. Springer-Verlag, New York (1987)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hadwin, D., Shen, J., Wu, W., Yuan, W.: Relative commutant of an unbounded operator affiliated with a finite von Neumann algebra. J. Oper. Theory 75(1), 209–223 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jorgensen, P.E.T., Ólafsson, G.: Unitary representations and Osterwalder-Schrader duality. In: The mathematical Legacy of Harish-Chandra (Baltimore, MD, : Proceeding of Symposium Pure Math., vol. 68, Amer. Math. Soc. Providence, RI Vol. 2000, pp. 333–401 (1998)Google Scholar
  14. 14.
    Jørgensen, P.E.T.: Essential self-adjointness of semibounded operators. Math. Ann. 237(2), 187–192 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jørgensen, P.E.T.: Unbounded operators: perturbations and commutativity problems. J. Funct. Anal. 39(3), 281–307 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jorgensen, P.E.T.: Essential self-adjointness of the graph-Laplacian. J. Math. Phys. 49(7), 073510–073533 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jorgensen, P.E.T., Pearse, E.P.J.: A Hilbert space approach to effective resistance metric. Complex Anal. Oper. Theory 4(4), 975–1013 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jorgensen, P.E.T., Pearse, E.P.J.: Gel’ fand triples and boundaries of infinite networks. N. Y. J. Math. 17, 745–781 (2011)zbMATHGoogle Scholar
  19. 19.
    Jorgensen, P.E.T., Pearse, E.P.J.: Spectral reciprocity and matrix representations of unbounded operators. J. Funct. Anal. 261(3), 749–776 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jorgensen, P.E.T., Pearse, E.P.J.: A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks. Israel J. Math. 196(1), 113–160 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jorgensen, P.E.T., Pearse, E.P.J.: Spectral comparisons between networks with different conductance functions. J. Oper. Theory 72(1), 71–86 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jorgensen, P.E.T., Pearse, E.P.J.: Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks. Complex Anal. Oper. Theory 10(7), 1535–1550 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jorgensen, P.E.T., Pearse, E.P.J.: Symmetric pairs of unbounded operators in Hilbert space, and their applications in mathematical physics. Math. Phys. Anal. Geom. 20(2), 14 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jorgensen, P., Pedersen, S., Tian, F.: Restrictions and extensions of semibounded operators. Complex Anal. Oper. Theory 8(3), 591–663 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jorgensen, P., Tian, F.: Infinite networks and variation of conductance functions in discrete laplacians. J. Math. Phys. 56(4), 043506 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kempf, A., Chatwin-Davies, A., Martin, R.T.W.: A fully covariant information-theoretic ultraviolet cutoff for scalar fields in expanding Friedmann Robertson Walker spacetimes. J. Math. Phys. 54(2), 022301–022322 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kreĭn, M.G., Langer, H.: On some extension problems which are closely connected with the theory of Hermitian operators in a space \(\varPi _{\kappa }\). III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part II. Beiträge Anal. 1980(15), 27–45 (1981)MathSciNetGoogle Scholar
  28. 28.
    Kreĭn, M.G.: On Hermitian operators with directed functionals. Akad. Nauk Ukrain. RSR. Zbirnik Prac’ Inst. Mat. 1948(10), 83–106 (1948)MathSciNetGoogle Scholar
  29. 29.
    Kreĭn, M.G.: The fundamental propositions of the theory of representations of Hermitian operators with deficiency index \((m, m)\). Ukrain. Mat. Žurnal 1(2), 3–66 (1949)MathSciNetGoogle Scholar
  30. 30.
    Martin, R.T.W.: Extensions of symmetric operators I: the inner characteristic function case. Concr. Oper. 2, 53–97 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Much, A.: Self-adjointness of deformed unbounded operators. J. Math. Phys. 56(9), 093501–093513 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nathanson, E.S., Jørgensen, P.E.T.: A global solution to the Schrödinger equation: from Henstock to Feynman. J. Math. Phys. 56(9), 092102–092115 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Park, D.K.: Green’s-function approach to two- and three-dimensional delta-function potentials and application to the spin-\(1/2\) Aharonov-Bohm problem. J. Math. Phys. 36(10), 5453–5464 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Powers, R.T.: Resistance inequalities for the isotropic Heisenberg ferromagnet. J. Math. Phys. 17(10), 1910–1918 (1976)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Reed, Michael, Simon, Barry: Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York-London (1972)zbMATHGoogle Scholar
  36. 36.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)zbMATHGoogle Scholar
  37. 37.
    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)zbMATHGoogle Scholar
  38. 38.
    Schmüdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, vol. 265. Springer, Dordrecht (2012)CrossRefzbMATHGoogle Scholar
  39. 39.
    Stone, M.H.: On unbounded operators in Hilbert space. J. Indian Math. Soc. (N.S.) 15, 155–192 (1952)MathSciNetzbMATHGoogle Scholar
  40. 40.
    von Neumann, J.: Über Funktionen von Funktionaloperatoren. Ann. Math. 32(2), 191–226 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Voiculescu, D.-V.: Free probability for pairs of faces I. Commun. Math. Phys. 332(3), 955–980 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhang, F. (ed.): The Schur Complement and Its Applications, Numerical Methods and Algorithms, vol. 4. Springer-Verlag, New York (2005)Google Scholar

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© 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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