Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

Riečanová, Zdenka; Zając, Michał; Pulmannová, Sylvia

doi:10.1016/s0034-4877(12)60009-3

Cited by 26 publications

(13 citation statements)

References 5 publications

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“…The situation with unbounded operators is from the algebraic point of view more complicated than one with bounded operators, see e.g. [9,10,13,14], where the structure of the generalized effect algebra V(H) of all positive linear operators and its properties was described.…”

Section: Introductionmentioning

confidence: 99%

On Bilinear Forms from the Point of View of Generalized Effect Algebras

Dvurečenskij

Janda

2013

Found Phys

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Section: Introductionmentioning

confidence: 99%

On Bilinear Forms from the Point of View of Generalized Effect Algebras

Dvurečenskij

Janda

2013

Found Phys

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“…We are going to prove that sets (2)- (7) and (9) and V * * (H) since for S p (H) it was proved in [14]. Thus we will get one element from every of the three classes introduced in Sect.…”

Section: Some Important Sub-generalized Effect Algebras Of the Operatmentioning

confidence: 92%

Considerable Sets of Linear Operators in Hilbert Spaces as Operator Generalized Effect Algebras

Paseka

Riečanová

2011

Found Phys

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We show that considerable sets of positive linear operators namely their extensions as closures, adjoints or Friedrichs positive self-adjoint extensions form operator (generalized) effect algebras. Moreover, in these cases the partial effect algebraic operation of two operators coincides with usual sum of operators in complex Hilbert spaces whenever it is defined. These sets include also unbounded operators which play important role of observables (e.g., momentum and position) in the mathematical formulation of quantum mechanics.

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“…Then for any D ∈ D, D = H, G D (H) forms a summability block of V D (H). Note that sub-generalized effect algebras G D (H) are also compatibility blocks (see [14], hence in this case, compatibility and summability blocks coincide. Theorem 7.…”

Section: Examplementioning

confidence: 99%

“…their isomorphism with sub-effect algebras of the standard effect algebra E(H) mentioned above) have been studied. It was proved in [14] that the set V D (H) of all positive linear operators in an infinite-dimensional complex Hilbert space H with partially defined sum of operators (which coincides with the usual sum) restricted to the common domains of operators forms a generalized effect algebra. This generalized effect algebra V D (H) is a union of sub-generalized effect algebras of maximal subsets of pairwise summable operators.…”

Section: Introduction and Some Basic Definitionsmentioning

confidence: 99%

Maximal Subsets of Pairwise Summable Elements in Generalized Effect Algebras

Riečanová

Janda

2013

Acta Polytech

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Abstract. We show that in any generalized effect algebra (G; ⊕, 0) a maximal pairwise summable subset is a sub-generalized effect algebra of (G; ⊕, 0), called a summability block. If G is lattice ordered, then every summability block in G is a generalized MV-effect algebra. Moreover, if every element of G has an infinite isotropic index, then G is covered by its summability blocks, which are generalized MV-effect algebras in the case that G is lattice ordered. We also present the relations between summability blocks and compatibility blocks of G. Counterexamples, to obtain the required contradictions in some cases, are given.Keywords: (generalized) effect algebra, MV-effect algebra, summability block, compatibility block, linear operators in Hilbert spaces.

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Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

Cited by 26 publications

References 5 publications

On Bilinear Forms from the Point of View of Generalized Effect Algebras

On Bilinear Forms from the Point of View of Generalized Effect Algebras

Considerable Sets of Linear Operators in Hilbert Spaces as Operator Generalized Effect Algebras

Maximal Subsets of Pairwise Summable Elements in Generalized Effect Algebras

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