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
References
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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.
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7. Berkovits J., Fabry C., An extension of the topological degree in Hilbert space, Abstr. Appl. Anal. 2005, no. 6, 581–597.
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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.
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.
MarianiM. C., TweneboahO. K., VallesM. A., & BezdekP. (2019). Complex Gleason measures and the Nemytsky operator. Annales Mathematicae Silesianae, 33, 168-209. Retrieved from https://www.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
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
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
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
Mathematics and Statistics Department, Utah State University, USA United States
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