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Considerable sets of linear operators in Hilbert spaces as operator generalized effect algebras. (English) Zbl 1238.81009
The authors study numerous families of positive linear operators on Hilbert spaces and show that these form (generalized) effect algebras. In these cases, when the partial effect-algebraic sum is defined, it coincides with the ordinary sum of operators. The possibility of extension of this partial operation to unbounded operators is discussed and related open problems are formulated.

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06C15 Complemented lattices, orthocomplemented lattices and posets
47B65 Positive linear operators and order-bounded operators
Full Text: DOI
[1] Arlinskiĭ, Yu., Tsekanovskiĭ, E.: On von Neumann’s problem in extension theory of nonnegative operators. Proc. Am. Math. Soc. 131, 3143–3154 (2003) · Zbl 1035.47006
[2] Bender, C.M., Boettcher, S.: Real spectra in non-hermitian hamiltonians having $\(\backslash\)mathcal{P}\(\backslash\)mathcal{T}$ symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998) · Zbl 0947.81018
[3] Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics, 2nd edn. Springer, Berlin (2008) · Zbl 1163.47060
[4] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000) · Zbl 0987.81005
[5] Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994) · Zbl 1213.06004
[6] Foulis, D.J.: Effects, observables, states, and symmetries in physics. Found. Phys. 37, 1421–1446 (2007) · Zbl 1129.81303
[7] Hedlíková, J., Pulmannová, S.: Generalized difference posets and orthoalgebras. Acta Math. Univ. Comen. LXV, 247–279 (1996) · Zbl 0922.06002
[8] Kalmbach, G., Riečanová, Z.: An axiomatization for abelian relative inverses. Demonstr. Math. 27, 769–780 (1994) · Zbl 0826.08002
[9] Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994)
[10] Paseka, J.: $\(\backslash\)mathcal{P}\(\backslash\)mathcal{T}$ -symmetry in (generalized) effect algebras. Int. J. Theor. Phys. 50, 1198–1205 (2011). doi: 10.1007/s10773-010-0594-9 · Zbl 1257.03094
[11] Polakovič, M., Riečanová, Z.: Generalized effect algebras of positive operators densely defined on Hilbert space. Int. J. Theor. Phys. 50, 1167–1174 (2011). doi: 10.1007/s10773-010-0458-3 · Zbl 1237.81009
[12] Reed, M., Simon, B.: Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness. Academic Press, New York (1975) · Zbl 0308.47002
[13] Riečanová, Z.: Subalgebras, intervals and central elements of generalized effect algebras. Int. J. Theor. Phys. 38, 3209–3220 (1999) · Zbl 0963.03087
[14] Riečanová, Z.: Effect algebras of positive self-adjoint operators densely defined on Hilbert spaces. Acta Polytech. (Proceedings of the 7-th DI Microconference Analytic and Algebraic Methods VII, Prague, March 2011, ed. V. Jakubský and M. Znojil). to appear
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