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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
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