Complex Gleason measures and the Nemytsky operator



Abstract

This work is devoted to the generalization of previous results on Gleason measures to complex Gleason measures. We develop a functional calculus for complex measures in relation to the Nemytsky operator. Furthermore we present and discuss the interpretation of our results with applications in the field of quantum mechanics. Some concrete examples and further extensions of several theorems are also presented.


Keywords

complex Gleason measures; Nemytsky operator; quantum mechanics

1. Aarnes J.F., Physical states on a C*-algebra, Acta Math. 122 (1969), 161–172.
2. Aarnes J.F., Quasi-states on C*-algebras, Trans. Amer. Math. Soc. 149 (1970), 601–625.
3. Alvarez J., Eydenberg M., Mariani M.C., The Nemytsky operator on vector valued measures, Preprint.
4. Alvarez J., Mariani M.C., Extensions of the Nemytsky operator: distributional solutions of nonlinear problems, J. Math. Anal. Appl. 338 (2008), no. 1, 588–598.
5. Amster P., Cassinelli M., Mariani M.C., Rial D.F., Existence and regularity of weak solutions to the prescribed mean curvature equation for a nonparametric surface, Abstr. Appl. Anal. 4 (1999), no. 1, 61–69.
6. Benyamini Y., Lindenstrauss J., Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society, Providence, 2000.
7. Berkovits J., Fabry C., An extension of the topological degree in Hilbert space, Abstr. Appl. Anal. 2005, no. 6, 581–597.
8. Berkovits J., Mawhin J., Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5041–5055.
9. Blank J., Exner P., Havlíček M., Hilbert Space Operators in Quantum Physics, Second edition, Springer, New York, 2008.
10. Blum K., Density Matrix Theory and Applications, Third edition, Springer, Berlin–Heidelberg, 2012.
11. Bunce L.J., Wright J.D. Maitland, The Mackey-Gleason problem, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 288–293.
12. Chevalier G., Dvurečenskij A., Svozil K., Piron's and Bell's geometric lemmas and Gleason's theorem, Found. Phys. 30 (2000), no. 10, 1737–1755.
13. Cohen-Tannoudji C., Diu B., Laloë F., Quantum Mechanics, Hermann and John Wiley & Sons, New York, 1977.
14. Cotlar M., Cignoli R., Nociones de Espacios Normados, Editorial Universitaria de Buenos Aires, Buenos Aires, 1971.
15. De Nápoli P., Mariani M.C., Some remarks on Gleason measures, Studia Math. 179 (2007), no. 2, 99–115.
16. Dinculeanu N., Vector Measures, Pergamon Press, Berlin, 1967.
17. Dunford N., Schwartz J., Linear Operators. I. General Theory, Interscience Publishers, New York–London, 1958.
18. Gaines R.E., Mawhin J.L., Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin–New York, 1977.
19. Gleason A.M., Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885–893.
20. Gunson J., Physical states on quantum logics. I, Ann. Inst. H. Poincaré Sect. A (N.S.) 17 (1972), 295–311.
21. Hemmick D.L., Hidden Variables and Nonlocality in Quantum Mechanics, Ph.D. thesis, 1996. Available at https://arxiv.org/abs/quant-ph/0412011v1.
22. Krasnosel'skii A.M., Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan Co., New York, 1964.
23. Krasnosel'skii A.M., Mawhin J., The index at infinity of some twice degenerate compact vector fields, Discrete Contin. Dynam. Systems 1 (1995), no. 2, 207–216.
24. Latif A., Banach contraction principle and its generalizations, in: Almezel S., Ansari Q.H., Khamsi M.A. (Eds.), Topics in Fixed Point Theory, Springer, Cham, 2014, pp. 33–64.
25. Maeda S., Probability measures on projections in von Neumann algebras, Rev. Math. Phys. 1 (1989), no. 2–3, 235–290.
26. Mawhin J., Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conf. Ser. in Math., 40, Amer. Math. Soc., Providence, 1979.
27. Mawhin J., Topological degree and boundary value problems for nonlinear differential equations, in: Furi M., Zecca P. (Eds.), Topological Methods for Ordinary Differential Equations, Lecture Notes in Math., 1537, Springer-Verlag, Berlin, 1993, pp. 74–142.
28. Messiah A., Quantum Mechanics, John Wiley & Sons, New York, 1958.
29. Morita T., Sasaki T., Tsutsui I., Complex probability measure and Aharonov's weak value, Prog. Theor. Exp. Phys. 2013, no. 5, 053A02, 11 pp.
30. Prugovečki E., Quantum Mechanics in Hilbert Spaces, Second edition, Academic Press, New York–London, 1981.
31. Richman F., Bridges D., A constructive proof of Gleason's theorem, J. Funct. Anal. 162 (1999), no. 2, 287–312.
32. Rieffel M.A., The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968), 466–487.
33. Riesz F., Sz.-Nagy B., Functional Analysis, Frederick Ungar Publishing Co., New York, 1955.
34. Ringrose J.R., Compact Non-self-adjoint Operators, Van Nostrand Reinhold Co., London, 1971.
35. Rudin W., Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, 1987.
36. Sherstnev A.N., The representation of measures that are de_ned on the orthoprojectors of Hilbert space by bilinear forms, Izv. Vysš. Učebn. Zaved. Matematika 1970 (1970), no. 9 (100), 90–97 (in Russian).
37. Vainberg M.M., Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco–London–Amsterdam, 1964.
38. von Neuman J., Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.
Download

Published : 2019-01-11


MarianiM. C., TweneboahO. K., VallesM. A., & BezdekP. (2019). Complex Gleason measures and the Nemytsky operator. Annales Mathematicae Silesianae, 33, 168-209. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13667

Maria C. Mariani  mcmariani@utep.edu
Department of Mathematical Sciences, University of Texas at El Paso, USA  United States
Osei K. Tweneboah 
Computational Science Program, University of Texas at El Paso, USA  United States
Miguel A. Valles 
Department of Mathematical Sciences, University of Texas at El Paso, USA  United States
Pavel Bezdek 
Mathematics and Statistics Department, Utah State University, USA  United States



The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.