Complex Gleason Measures and the Nemytsky Operator

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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.

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  • [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

  • [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 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] 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.

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