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A Hilbert space operator representation of abelian po-groups of bilinear forms. (English) Zbl 1329.81097
Summary: The existence of a non-trivial singular positive bilinear form [B. Simon, J. Funct. Anal. 28, 377–385 (1978; Zbl 0413.47029)] yields that on an infinite-dimensional complex Hilbert space $$\mathcal H$$ the set of bilinear forms $$\mathcal F(\mathcal H)$$ is richer than the set of linear operators $$\mathcal V(\mathcal H)$$. We show that there exists an structure preserving embedding of partially ordered groups from the abelian po-group $$\mathcal S_D(\mathcal H)$$ of symmetric bilinear forms with a fixed domain $$D$$ on a Hilbert space $$\mathcal H$$ into the po-group of linear symmetric operators on a dense linear subspace of an infinite dimensional complex Hilbert space $$l_2(M)$$. Moreover, if we restrict ourselves to the positive parts of the above mentioned po-groups, we can embed positive bilinear forms into corresponding positive linear operators.
##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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##### References:
 [1] Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics, 2nd ed. Springer, Berlin (2008) · Zbl 1268.81014 [2] Chajda, I; Paseka, J; Lei, Q, On realization of partially ordered abelian groups, Inter. J. Theor. Phys., 52, 2028-2037, (2013) · Zbl 1270.81011 [3] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academy Publication, Dordrecht/Ister Science (2000) · Zbl 0987.81005 [4] Dvurečenskij, A; Janda, J, On bilinear forms from the point of view of generalized effect algebras, Found. Phys., 43, 1136-1152, (2013) · Zbl 1285.81003 [5] Foulis, DJ; Bennett, MK, Effect algebras and unsharp quantum logics, Found. Phys., 24, 1331-1352, (1994) · Zbl 1213.06004 [6] Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer Verlag, Berlin/Heidelberg/New York (1976) · Zbl 0342.47009 [7] Paseka, J, On realization of generalized effect algebras, Rep. Math. Phys., 70, 375-384, (2012) · Zbl 1296.03038 [8] 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 [9] Polakovič, M; Riečanová, Z, Generalized effect algebras of positive operators densely defined on Hilbert spaces, Inter. J. Theor. Phys., 50, 1167-1174, (2011) · Zbl 1237.81009 [10] Pulmannová, S, Representations of MV-algebras by Hilbert-space effects, Inter. J. Theor. Phys., 52, 2163-2170, (2013) · Zbl 1270.06006 [11] 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 [12] Riečanová, Z; Zajac, M, Hilbert space effect-representations of effect algebras, Rep. Math. Phys., 70, 283-290, (2012) · Zbl 1268.81014 [13] Simon, B, A canonical decomposition for quadratic forms with applications for monotone convergence theorems, J. Funct. Anal., 28, 377-385, (1978) · Zbl 0413.47029
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