zbMATH — the first resource for mathematics

On bilinear forms from the point of view of generalized effect algebras. (English) Zbl 1285.81003
The authors show that various families of positive bilinear forms on a Hilbert space might be organized as a generalized effect algebra and clarify which of considered subfamilies are substructures. Moreover, they study monotone Dedekind downwards and upwards \(\sigma\)-completeness of these effect algebras and present an example that we can obtain another result if we use the usual partial order of positive bilinear forms instead of that induced by the generalized effect algebra structure.

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G25 Other algebras related to logic
03G12 Quantum logic
Full Text: DOI arXiv
[1] Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics, 2nd edn. Springer, Berlin (2008) · Zbl 1163.47060
[2] Dvurečenskij, A.: Gleason’s Theorem and Its Applications. Mathematics and Its Applications, vol. 60. Kluwer Acad. Publ, Dordrecht/Ister Science, Bratislava (1993) · Zbl 1250.81015
[3] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht/Ister Science, Bratislava (2000)
[4] Foulis, D.J.; Bennett, M.K., Effect algebras and unsharp quantum logics, Found. Phys., 24, 1331-1352, (1994) · Zbl 1213.06004
[5] Halmos, P.R.: Introduction to Hilbert Space and the Theory of Spectral Multiplicity, 2nd edn. Chelsea Publ. Co., New York (1957) · Zbl 0079.12404
[6] Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976) · Zbl 0342.47009
[7] Kôpka, F.; Chovanec, F., D-posets, Math. Slovaca, 44, 21-34, (1994) · Zbl 0789.03048
[8] Lugovaya, G.D., Bilinear forms defining measures on projectors, Izv. Vysš. Učebn. Zaved., Mat., 249, 88, (1983) · Zbl 0511.46061
[9] Paseka, J.; Riečanová, Z., Considerable sets of linear operators in Hilbert spaces as operator generalized effect algebras, Found. Phys., 41, 1634-1647, (2011) · Zbl 1238.81009
[10] Polakovič, M.; Riečanová, Z., Generalized effect algebras of positive operators densely defined on Hilbert spaces, Int. J. Theor. Phys., 50, 1167-1174, (2011) · Zbl 1237.81009
[11] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. I: Functional Analysis. Academic Press, New York (1972) · Zbl 0242.46001
[12] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-adjointness. Academic Press, San Diego (1975) · Zbl 0308.47002
[13] Riečanová, Z.; Zajac, M.; Pulmannová, S., Effect algebras of positive linear operators densely defined on Hilbert spaces, Rep. Math. Phys., 68, 261-270, (2011) · Zbl 1250.81015
[14] Riečanová, Z.; Zajac, M., Hilbert space effect-representations of effect algebras, Rep. Math. Phys., 70, 283-290, (2012) · Zbl 1268.81014
[15] Simon, B., A canonical decomposition for quadratic forms with applications for monotone convergence theorems, J. Funct. Anal., 28, 377-385, (1978) · Zbl 0413.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.