# zbMATH — the first resource for mathematics

Weakly ordered partial commutative group of self-adjoint linear operators densely defined on Hilbert space. (English) Zbl 1302.47062
The set of all bounded positive linear operators in a complex Hilbert space $$H$$ dominated by identity operator equiped with partially defined sum possess algebraic structure, which was a prototype of the so-called effect algebras, introduced by D. J. Foulis and M. K. Bennett [Found. Phys. 24, No. 10, 1331–1352 (1994; Zbl 1213.06004)]. Recently, new examples of (generalized) effect algebras, including also unbounded positive linear operators densely defined in $$H$$, were given by M. Polakovič and Z. Riečanová [Int. J. Theor. Phys. 50, No. 4, 1167–1174 (2011; Zbl 1237.81009)], Z. Riečanová et al. [Rep. Math. Phys. 68, No. 3, 261–270 (2011; Zbl 1250.81015)] and J. Paseka and Z. Riečanová [Found. Phys. 41, No. 10, 1634–1647 (2011; Zbl 1238.81009)]. Simultaneously, J. Paseka and J. Janda [“More on $$\mathcal{PT}$$-symmetry in (generalized) effect algebras and partial groups”, Acta Polytech., Pr. ČVUT Praha 51, No. 4, 65–72 (2011), https://ojs.cvut.cz/ojs/index.php/ap/article/view/1408/1240] presented a structure of weakly ordered commutative group (wop-group, for short) of all linear operators on $$H$$ with the usual sum restricted for unbounded operators only on pairs with the same domains. The referred paper is a continuation in this direction. The author proved that the set of all self-adjoint operators with partial operation defined as by J. Paseka and Z. Riečanová [Found. Phys. 41, No. 10, 1634–1647 (2011; Zbl 1238.81009)] for positive operators, forms a wop-group. The author shows some properties of these wop-groups and their relation to generalized effect algebras.
##### MSC:
 47D03 Groups and semigroups of linear operators 06F15 Ordered groups 03G25 Other algebras related to logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 08A55 Partial algebras
Full Text:
##### References:
 [1] BLANK, J.-EXNER, P.-HAVLÍČEK,M.: Hilbert Space Operators in Quantum Physics , Springer-Verlag, Berlin, 2008. · Zbl 1163.47060 [2] HEDLÍKOVÁ, J.-PULMANNOVÁ, S.: Generalized difference posets and orthoalgebras, Acta Math. Univ. Comenianae 45 (1996), 247-279. · Zbl 0922.06002 · emis:journals/AMUC/_vol-65/_no_2/_hedliko/hedlikov.html [3] DVUREČENSKIJ, A.-PULMANNOVÁ, S.: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht/Ister Science, Bratislava, 2000. · Zbl 0987.81005 [4] FOULIS, D. J.-BENNETT, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331-1352. · Zbl 1213.06004 · doi:10.1007/BF02283036 [5] PASEKA, J.: PT -symmetry in (generalized) effect algebras, Internat. J. Theoret. Phys. 50 (2011), 1198-1205. · Zbl 1257.03094 · doi:10.1007/s10773-010-0594-9 [6] PASEKA, J.-JANDA, J.: More on PT -symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65-72. [7] PASEKA, J.-RIEČANOVÁ, Z.: Considerable sets of linear operators in Hilbert spaces as generalized effect algebras. Found. Phys. 41 (2011), 1634-1647. · Zbl 1238.81009 · doi:10.1007/s10701-011-9573-0 · gateway.webofknowledge.com [8] POLAKOVIĆ, M.-RIEĆANOVÁ , Z.: Generalized effect algebras of positive operators densely defined on Hilbert spaces, Internat. J. Theoret. Phys. 50 (2011), 1167-1174. · Zbl 1237.81009 · doi:10.1007/s10773-010-0458-3 [9] RIEĆANOVÁ, Z.-ZAJAC, M.-PULMANNOVÁ, S.: Effect algebras of positive opera- tors densely defined on Hilbert spaces, Rep. Math. Phys., 2011 (accepted). [10] RIEĆANOVÁ, Z.: Effect algebras of positive self-adjoint operators densely defined on Hilbert spaces, Acta Polytech. 51 (2011), 78-82. [11] RIEĆANOVÁ, Z.-ZAJAC, M.: Extensions of effect algebra operations, Acta Polytech. 51 (2011), 73-77.
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.