Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

2011 ◽  
Vol 68 (3) ◽  
pp. 261-270 ◽  
Author(s):  
Z. Riečanová ◽  
M. Zajac ◽  
S. Pulmannová
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Effect Algebras ◽  
Densely Defined

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.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

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 UNBOUNDED B-FREDHOLM OPERATORS ON HILBERT SPACES

2008 ◽  
Vol 51 (2) ◽  
pp. 285-296 ◽  
Author(s):  
M. Berkani ◽  
N. Castro-González
Keyword(s):  
Fredholm Operator ◽  
Hilbert Spaces ◽  
Closed Operator ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Spectral Mapping ◽  
Nilpotent Operator ◽  
Densely Defined ◽  
Closed Linear Operators

AbstractThis paper is concerned with the study of a class of closed linear operators densely defined on a Hilbert space $H$ and called B-Fredholm operators. We characterize a B-Fredholm operator as the direct sum of a Fredholm closed operator and a bounded nilpotent operator. The notion of an index of a B-Fredholm operator is introduced and a characterization of B-Fredholm operators with index $0$ is given in terms of the sum of a Drazin closed operator and a finite-rank operator. We analyse the properties of the powers $T^m$ of a closed B-Fredholm operator and we establish a spectral mapping theorem.

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 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.

The Spectral Resolution of Some Non-Selfadjoint Partial Differential Operators

1975 ◽  
Vol 27 (6) ◽  
pp. 1316-1322
Author(s):  
Gustavus E. Huige
Keyword(s):  
Spectral Resolution ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Linear Operators ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Densely Defined

Let ffl and be Hilbert spaces, the set of all densely defined linear operators from to , and its subset of bounded ones. Let and σ(T) denote the adjoint, range, domain, closure and spectrum of T respectively. Ri(z) will denote the resolvent (z — Ti)-1.

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