Further Remarks on an Order for Quantum Observables
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Jānis Cīrulis
Abstract
S. Gudder and, later, S. Pulmanová and E. Vinceková, have studied in two recent papers a certain ordering of bounded self-adjoint operators on a Hilbert space. We present some further results on this ordering and show that some structure theorems of the ordered set of operators can be obtained in a more abstract setting of posets having the upper bound property and equipped with a certain orthogonality relation.
References
[1] ANTEZANA, J.-CANO, C. et al: A note on the star order in Hilbert spaces, Linear Multilinear Algebra 58 (2010), 1037-1051.10.1080/03081080903227104Search in Google Scholar
[2] CHAJDA, I.-KOLAŘĺK, M.: Nearlattices, Discrete Math. 308 (2008), 4906-4913.10.1016/j.disc.2007.09.009Search in Google Scholar
[3] CĪRULIS, J.: Subtractive nearsemilattices, Proc. Latv. Acad. Sci. Sect. B Nat. Exact Appl. Sci. 52 (1998), 228-233.Search in Google Scholar
[4] CĪRULIS, J.: Knowledge representation in extended Pawlak’s information systems: algebraic aspects. In: FOIKS 2002 (T. Eiter, K.-D. Schewe, eds.). Lecture Notes in Comput. Sci. 2284, Springer-Verlag, Berlin, 2002, pp. 250-267.Search in Google Scholar
[5] CĪRULIS, J.: Knowledge representation systems and skew nearlattices. In: Contributions to General Algebra 14 (I. Chajda, et al, eds.), Verlag Johannes Heyn, Klagenfurt, 2004, pp. 43-51.Search in Google Scholar
[6] CĪRULIS, J: Skew nearlattices: some structure and representation theorems. In: Contributions to General Algebra 19 (I. Chajda, et al, eds.), Verlag Johannes Heyn, Klagenfurt, 2010, pp. 33-44.10.1007/s11083-010-9152-6Search in Google Scholar
[7] CĪRULIS, J.: Subtraction-like operations in nearsemilattices, Demonstratio Math. 43 (2010), 725-738.Search in Google Scholar
[8] CORNISH, W. H.: Conversion of nearlattices into implicative BCK-algebras, Math. Semin. Notes Kobe Univ. 10 (1982), 1-8.Search in Google Scholar
[9] CORNISH, W. H.-NOOR, A. S. A.: Standard elements in a nearlattice, Bull. Aust. Math. Soc. 26 (1982), 185-213.10.1017/S0004972700005700Search in Google Scholar
[10] DRAZIN, M. F.: Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139-141.10.1090/S0002-9904-1978-14442-5Search in Google Scholar
[11] DVUREČENSKIJ, A.-PULMANNOVÁ, S.: New Trends in Quantum Structures, Kluwer Acad. Publ./Ister Sci., Dordrecht/Bratislava, 2000.10.1007/978-94-017-2422-7Search in Google Scholar
[12] DU, H.-DOU, Y.: A spectral representation of the infimum of selfadjoint operators in the logic order, Acta Math. Sinica (Chin. Ser.) 52 (2009), 1141-1146 (Chinese).Search in Google Scholar
[13] GUDDER, S: An order for quantum observables, Math. Slovaca 56 (2006), 573-589.Search in Google Scholar
[14] JANOWITZ, M.F.: A note on generalized orthomodular lattices, J. Natur. Sci. Math. 8 (1968), 89-94.Search in Google Scholar
[15] LIU,W.-WU, J.: A representation theorem of infimum of bounded observables, J.Math. Phys. 49 (2008), Article No. 073521, 5 pp.Search in Google Scholar
[16] LIU, W.-WU, J.: A supremumum of bounded quantum observables, J. Math. Phys. 50 (2009), Article No. 083513, 4 pp.Search in Google Scholar
[17] MAYET-IPPOLITO, A.: Generalized orthomodular posets, Demonstratio Math. 24 (1991), 263-274.Search in Google Scholar
[18] NOOR, A. S. A.-CORNISH,W. H.: Multipliers on a nearlattice, Comment. Math. Univ. Carolin. 27 (1986), 815-827.Search in Google Scholar
[19] PULMANNOVÁ, S.-VINCEKOVÁ, E.: Remarks on the order for quantum observables, Math. Slovaca 57 (2007), 589-600.10.2478/s12175-007-0048-xSearch in Google Scholar
[20] SHEN, J,-WU, J.: Spectral representation of infimum of bounded quantum observables, J. Math. Phys. 50 (2009), Article No. 1135014, 4 pp.Search in Google Scholar
[21] XU, X.-DU, H.-FANG, X.: An explicit expression of supremum of bounded quantum observables, J. Math. Phys. 50 (2009), Article No. 033502, 9 pp. Search in Google Scholar
Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Radius, Diameter and the Degree Sequence of a Graph
- On Primary Ideals in Posets
- Characterization of the Set of Regular Elements in Ordered Semigroups
- Tame Automorphisms with Multidegrees in the Form of Arithmetic Progressions
- A Result Concerning Additive Mappings in Semiprime Rings
- Characterizing Jordan Derivations of Matrix Rings Through Zero Products
- Existence Results for Impulsive Nonlinear Fractional Differential Equations With Nonlocal Boundary Conditions
- The Radon-Nikodym Property and the Limit Average Range
- A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting
- On Booth Lemniscate and Hadamard Product of Analytic Functions
- Recursion Formulas for Srivastava Hypergeometric Functions
- Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments
- Singular Degenerate Differential Operators and Applications
- The Interior Euler-Lagrange Operator in Field Theory
- On Selections of Set-Valued Maps Satisfying Some Inclusions in a Single Variable
- The Family F of Permutations of ℕ
- Summation Process of Positive Linear Operators in Two-Dimensional Weighted Spaces
- On Iλ-Statistical Convergence in Locally Solid Riesz Spaces
- Some Norm one Functions of the Volterra Operator
- Some Results on Absolute Retractivity of the Fixed Points Set of KS-Multifunctions
- Convexity in the Khalimsky Plane
- Natural Boundary Conditions in Geometric Calculus of Variations
- Exponential Inequalities for Bounded Random Variables
- Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences
- Second Order Riemannian Mechanics
- Further Remarks on an Order for Quantum Observables
