On a generalized concept of order relations on B(H)

2018 ◽  
Vol 68 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Gregor Dolinar ◽  
Janko Marovt
Keyword(s):  
Hilbert Space ◽  
Partial Order ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Order Relations ◽  
The Right ◽  
Minus Partial Order

AbstractLetHbe a Hilbert space andB(H) the set of all bounded linear operators onH. In the paper we consider the generalized concept of order relations onB(H) which was proposed by Šemrl and which covers the star partial order, the left-star partial order, the right-star partial order, and the minus partial order. We also connect this concept with the sharp partial order.

WEIGHTED CORE–EP INVERSE AND WEIGHTED CORE–EP PRE-ORDERS IN A -ALGEBRA

2019 ◽  
pp. 1-35 ◽  
Author(s):  
DIJANA MOSIĆ
Keyword(s):  
Partial Order ◽  
Hilbert Spaces ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Core ◽  
Invertible Elements ◽  
Minus Partial Order

We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$ -algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$ -Drazin invertible elements of a $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

scholarly journals Proving the Equality of the Spaces Q_b^r (A),Q_b^l (A) and BL(X) where X is a Complex Banach Space

2020 ◽  
pp. 127-131
Author(s):  
Mohammed Th. Al-Neima ◽  
Amir A. Mohammed
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Algebra ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Right ◽  
Norm Ideal

Cabrera and Mohammed proved that the right and left bounded algebras of quotients  and  of norm ideal  on a Hilbert space  are equal to  Banach algebra of all bounded linear operators on . In this paper, we prove that  where  is a norm ideal on a complex Banach space .

Partial orders on the power sets of Baer rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050011 ◽  
Author(s):  
B. Ungor ◽  
S. Halicioglu ◽  
A. Harmanci ◽  
J. Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Baer Ring ◽  
Power Set ◽  
Von Neumann Regular ◽  
Baer Rings ◽  
The Right

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

Free E0-Semigroups

1995 ◽  
Vol 47 (4) ◽  
pp. 744-785 ◽  
Author(s):  
Neal J. Fowler
Keyword(s):  
Hilbert Space ◽  
Free Product ◽  
Continuous Semigroup ◽  
Fock Space ◽  
Linear Operators ◽  
Free Flow ◽  
Strongly Continuous Semigroup ◽  
Bounded Linear Operators ◽  
Bounded Linear

AbstractGiven a strongly continuous semigroup of isometries ∪ acting on a Hilbert space ℋ, we construct an E0-semigroup α∪, the free E0-semigroup over ∪, acting on the algebra of all bounded linear operators on full Fock space over ℋ. We show how the semigroup αU⊗V can be regarded as the free product of α∪ and αV. In the case where U is pure of multiplicity n, the semigroup au, called the Free flow of rank n, is shown to be completely spatial with Arveson index +∞. We conclude that each of the free flows is cocycle conjugate to the CAR/CCR flow of rank +∞.

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 Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space

1988 ◽  
Vol 31 (1) ◽  
pp. 127-144 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Hilbert Space ◽  
Spectral Theory ◽  
Linear Operators ◽  
Coupled Systems ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Image Position

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operatorswhere λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

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.

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