scholarly journals Extensions of Effect Algebra Operations

10.14311/1410 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
Z. Riečanová ◽  
M. Zajac
Keyword(s):  
Effect Algebra ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Effect Algebras ◽  
Infinite Dimensional ◽  
Interval Effect ◽  
Generalized Effect Algebra ◽  
Algebraic Operations ◽  
Densely Defined ◽  
Generalized Effect Algebras

We study the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space. We equip this set with various effect algebraic operations making it a generalized effect algebra. Further, sub-generalized effect algebras and interval effect algebras with respect of these operations are investigated.

scholarly journals Weakly ordered partial commutative group of self-adjoint linear operators densely defined on Hilbert space

2011 ◽  
Vol 50 (1) ◽  
pp. 63-78
Author(s):  
Jiří Janda
Keyword(s):  
Hilbert Space ◽  
Algebraic Structure ◽  
Complex Hilbert Space ◽  
Commutative Group ◽  
Pt Symmetry ◽  
Linear Operators ◽  
Effect Algebras ◽  
Infinite Dimensional ◽  
Densely Defined ◽  
Generalized Effect Algebras

ABSTRACT We continue in a direction of describing an algebraic structure of linear operators on infinite-dimensional complex Hilbert space ℋ. In [Paseka, J.- -Janda, J.: More on PT-symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65-72] there is introduced the notion of a weakly ordered partial commutative group and showed that linear operators on H with restricted addition possess this structure. In our work, we are investigating the set of self-adjoint linear operators on H showing that with more restricted addition it also has the structure of a weakly ordered partial commutative group.

scholarly journals Effect Algebras of Positive Self-adjoint Operators Densely Defined on Hilbert Spaces

10.14311/1412 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
Z. Riečanová
Keyword(s):  
Hilbert Spaces ◽  
Binary Operation ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Algebraic Operation ◽  
Effect Algebras ◽  
Infinite Dimensional ◽  
Adjoint Operators ◽  
Densely Defined ◽  
Generalized Effect Algebras

We show that (generalized) effect algebras may be suitable very simple and natural algebraic structures for sets of (unbounded) positive self-adjoint linear operators densely defined on an infinite-dimensional complex Hilbert space. In these cases the effect algebraic operation, as a total or partially defined binary operation, coincides with the usual addition of operators in Hilbert spaces.

scholarly journals Intervals in Generalized Effect Algebras and their Sub-generalized Effect Algebras

10.14311/1817 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Zdenka Riečanová ◽  
Michal Zajac
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Generalized Effect Algebra ◽  
Generalized Effect Algebras

We consider subsets G of a generalized effect algebra E with 0∈G and such that every interval [0, q]G = [0, q]E ∩ G of G (q ∈ G , q ≠ 0) is a sub-effect algebra of the effect algebra [0, q]E. We give a condition on E and G under which every such G is a sub-generalized effect algebra of E.

scholarly journals Weakly Ordered A-Commutative Partial Groups of Linear Operators Densely Defined on Hilbert Space

10.14311/1807 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Jirí Janda
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Commutative Group ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Generalized Effect Algebra ◽  
Algebraic Description ◽  
Densely Defined

The notion of a generalized effect algebra is presented as a generalization of effect algebra for an algebraic description of the structure of the set of all positive linear operators densely defined on a Hilbert space with the usual sum of operators. The structure of the set of not only positive linear operators can be described with the notion of a weakly ordered partial commutative group (wop-group).Due to the non-constructive algebraic nature of the wop-group we introduce its stronger version called a weakly ordered partial a-commutative group (woa-group). We show that it also describes the structure of not only positive linear operators.

scholarly journals MAXIMAL SUBSETS OF PAIRWISE SUMMABLE ELEMENTS IN GENERALIZED EFFECT ALGEBRAS

2013 ◽  
Vol 53 (5) ◽  
pp. 457-461 ◽  
Author(s):  
Zdenka Riečanová ◽  
Jiří Janda
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Generalized Effect Algebra ◽  
Generalized Effect Algebras

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.

Some properties of D-weak operator topology

2020 ◽  
Vol 70 (3) ◽  
pp. 753-758
Author(s):  
Marcel Polakovič
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Linear Subspace ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Weak Operator Topology ◽  
Generalized Effect Algebra ◽  
Weak Operator ◽  
Dense Linear Subspace

AbstractLet 𝓖D(𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖D(𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖D(𝓗) is the whole 𝓖D(𝓗). The closure of the set of all unbounded elements of 𝓖D(𝓗) is also the set 𝓖D(𝓗). If Q is arbitrary unbounded element of 𝓖D(𝓗), it determines an interval in 𝓖D(𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

Operators with Compact Self-Commutator

1974 ◽  
Vol 26 (1) ◽  
pp. 115-120 ◽  
Author(s):  
Carl Pearcy ◽  
Norberto Salinas
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Open Problems ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Calkin Algebra ◽  
The Ideal

Let be a fixed separable, infinite dimensional complex Hilbert space, and let () denote the algebra of all (bounded, linear) operators on . The ideal of all compact operators on will be denoted by and the canonical quotient map from () onto the Calkin algebra ()/ will be denoted by π.Some open problems in the theory of extensions of C*-algebras (cf. [1]) have recently motivated an increasing interest in the class of all operators in () whose self-commuta tor is compact.

Weyl's theorem for the square of operator and perturbations

2015 ◽  
Vol 17 (05) ◽  
pp. 1450042
Author(s):  
Weijuan Shi ◽  
Xiaohong Cao
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
The Stability

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

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