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Complex Gleason Measures and the Nemytsky Operator

Open Access
|Jul 2019

References

  1. [1] Aarnes J.F., Physical states on a C* -algebra, Acta Math. 122 (1969), 161–172.10.1007/BF02392009
  2. [2] Aarnes J.F., Quasi-states on C* -algebras, Trans. Amer. Math. Soc. 149 (1970), 601–625.10.2307/1995417
  3. [3] Alvarez J., Eydenberg M., Mariani M.C., The Nemytsky operator on vector valued measures, Preprint.
  4. [4] Alvarez J., Mariani M.C., Extensions of the Nemytsky operator: distributional solutions of nonlinear problems, J. Math. Anal. Appl. 338 (2008), no. 1, 588–598.10.1016/j.jmaa.2007.05.026
  5. [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, 61–69.10.1155/S1085337599000019
  6. [6] Benyamini Y., Lindenstrauss J., Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society, Providence, 2000.10.1090/coll/048
  7. [7] Berkovits J., Fabry C., An extension of the topological degree in Hilbert space, Abstr. Appl. Anal. 2005, no. 6, 581–597.10.1155/AAA.2005.581
  8. [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, 5041–5055.10.1090/S0002-9947-01-02875-6
  9. [9] Blank J., Exner P., Havlíček M., Hilbert Space Operators in Quantum Physics, Second edition, Springer, New York, 2008.
  10. [10] Blum K., Density Matrix Theory and Applications, Third edition, Springer, Berlin–Heidelberg, 2012.10.1007/978-3-642-20561-3
  11. [11] Bunce L.J., Wright J.D. Maitland, The Mackey-Gleason problem, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 288–293.10.1090/S0273-0979-1992-00274-4
  12. [12] Chevalier G., Dvurečenskij A., Svozil K., Piron’s and Bell’s geometric lemmas and Gleason’s theorem, Found. Phys. 30 (2000), no. 10, 1737–1755.10.1023/A:1026458519154
  13. [13] Cohen-Tannoudji C., Diu B., Laloë F., Quantum Mechanics, Hermann and John Wiley & Sons, New York, 1977.
  14. [14] Cotlar M., Cignoli R., Nociones de Espacios Normados, Editorial Universitaria de Buenos Aires, Buenos Aires, 1971.
  15. [15] De Nápoli P., Mariani M.C., Some remarks on Gleason measures, Studia Math. 179 (2007), no. 2, 99–115.10.4064/sm179-2-1
  16. [16] Dinculeanu N., Vector Measures, Pergamon Press, Berlin, 1967.10.1016/B978-1-4831-9762-3.50004-4
  17. [17] Dunford N., Schwartz J., Linear Operators. I. General Theory, Interscience Publishers, New York–London, 1958.
  18. [18] Gaines R.E., Mawhin J.L., Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin–New York, 1977.10.1007/BFb0089537
  19. [19] Gleason A.M., Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885–893.10.1512/iumj.1957.6.56050
  20. [20] Gunson J., Physical states on quantum logics. I, Ann. Inst. H. Poincaré Sect. A (N.S.) 17 (1972), 295–311.
  21. [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. [22] Krasnosel’skii A.M., Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan Co., New York, 1964.
  23. [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, 207–216.10.3934/dcds.1995.1.207
  24. [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. 33–64.10.1007/978-3-319-01586-6_2
  25. [25] Maeda S., Probability measures on projections in von Neumann algebras, Rev. Math. Phys. 1 (1989), no. 2–3, 235–290.10.1142/S0129055X89000122
  26. [26] Mawhin J., Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conf. Ser. in Math., 40, Amer. Math. Soc., Providence, 1979.10.1090/cbms/040
  27. [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. 74–142.10.1007/BFb0085076
  28. [28] Messiah A., Quantum Mechanics, John Wiley & Sons, New York, 1958.
  29. [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.10.1093/ptep/ptt017
  30. [30] Prugovečki E., Quantum Mechanics in Hilbert Spaces, Second edition, Academic Press, New York–London, 1981.
  31. [31] Richman F., Bridges D., A constructive proof of Gleason’s theorem, J. Funct. Anal. 162 (1999), no. 2, 287–312.10.1006/jfan.1998.3372
  32. [32] Rieffel M.A., The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968), 466–487.10.1090/S0002-9947-1968-0222245-2
  33. [33] Riesz F., Sz.-Nagy B., Functional Analysis, Frederick Ungar Publishing Co., New York, 1955.
  34. [34] Ringrose J.R., Compact Non-self-adjoint Operators, Van Nostrand Reinhold Co., London, 1971.
  35. [35] Rudin W., Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, 1987.
  36. [36] Sherstnev A.N., The representation of measures that are defined on the orthoprojectors of Hilbert space by bilinear forms, Izv. Vysš. Učebn. Zaved. Matematika 1970 (1970), no. 9 (100), 90–97 (in Russian).
  37. [37] Vainberg M.M., Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco–London–Amsterdam, 1964.
  38. [38] von Neuman J., Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.
DOI: https://doi.org/10.2478/amsil-2018-0012 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107
Language: English
Page range: 168 - 209
Submitted on: Aug 31, 2017
Accepted on: Nov 28, 2018
Published on: Jul 18, 2019
Published by: University of Silesia in Katowice, Institute of Mathematics
In partnership with: Paradigm Publishing Services
Publication frequency: 2 times per year

© 2019 Maria C. Mariani, Osei K. Tweneboah, Miguel A. Valles, Pavel Bezdek, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.