On the extension of positive linear functionals

Muhamadiev, Ergashboy Mirzoevich; Diab, Adel T.

doi:10.1155/s0161171200001721

Cited by 3 publications

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References 2 publications

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“…The next result (which generalizes also a result of [12]), shows that it is possible to drop the condition that the sublinear operator S is monotone if the condition …”

Section: Extensions Of Positive Linear Operatorssupporting

confidence: 63%

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Existence and extensions of positive linear operators

Dăneţ

2008

Positivity

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In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators.In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators.In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. MathematicsSubject Classification (2000). Primary 46A22; Secondary 47B60, 47B65. Keywords. Mazur-Orlicz theorem, extensions of positive linear operators, sublinear operators, linear operators on ordered vector spaces. 90 N. Dȃneţ and R.-M. Dȃneţ Positivity T 1 (a) ≤ T 2 (a) for all a ∈ A. But, attention, a linear operator T :for all x 1 , x 2 ∈ X, and positively homogeneous, i.e., S(λx) = λS(x) for all x ∈ X and λ ≥ 0. The following famous theorem, called the HahnBanach existence theorem, is very well known. (For a proof see, for example, [11], p. 44.) Theorem 1.1. For every sublinear operator S : X −→ F there exists a linear operator L : X −→ F such that L ≤ S on X.

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“…The next result (which generalizes also a result of [12]), shows that it is possible to drop the condition that the sublinear operator S is monotone if the condition …”

Section: Extensions Of Positive Linear Operatorssupporting

confidence: 63%

“…The following result, which appears in [12] for F = R, shows such a situation. In our presentation it is a simple consequence of Theorem 2.4.…”

Section: Extensions Of Positive Linear Operatorsmentioning

confidence: 75%

Existence and extensions of positive linear operators

Dăneţ

2008

Positivity

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“…This is a special case of a much more general theorem. See [32] for a proof and related results. We will now prove a version of the Cauchy-Schwarz inequality for positive linear functionals.…”

Section: Example Supposementioning

confidence: 99%

$C^*$-algebras: The (Quantum) Path Less Traveled

Sekhon

2021

Preprint

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A Embedding B The Weak* Topology References Conventions • F denotes either R or C. • N denotes the set {1, 2, 3, ...} of natural numbers (excluding 0). • N 0 denotes the set {0, 1, 2, 3, ...} of natural numbers (including 0). • D(z 0 , r) denotes the open disk {z ∈ C | |z − z 0 | < r} in the complex plane. • D(z 0 , r) denotes the closed disk {z ∈ C | |z − z 0 | ≤ r} in the complex plane.• Inner products are assumed to be linear in the first argument and conjugate linear in the second.• H denotes a (real or complex) Hilbert space.• All algebras are assumed to be associative.

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“…The Hahn-Banach theorem has important implications for convex sets, and is the foundation for an effective treatment of optimization. It is also used to solve many problems in linear algebra, conic duality theory, the minimax theorem, piecewise approximation of convex functionals, extensions of positive linear functionals, and other results from modern control [1,5,8]. The Hahn-Banach theorem was proved by Hans Hahn in 1927 and later by Stefan Banach in 1929.…”

Section: Introductionmentioning

confidence: 99%

The Sandwich Theorem for Sublinear and Superlinear Functionals

Diab,

Nada,

Fearnley

2016

Preprint

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The Hahn-Banach theorem is an extension theorem for linear functionals which preserves certain properties. Specifically, if a linear functional is defined on a subspace of a real vector space which is dominated by a sublinear functional on the entire space, then this functional can be extended to a linear functional on the entire space which is still dominated by a sublinear functional. In this paper, we generalize this result to show that a linear functional defined on a subspace of a real vector space which is dominated by a sublinear functional and also dominates a superlinear functional on the entire space can be extended to a linear functional on the entire space which is also dominated by a sublinear functional and dominates a superlinear functional.

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On the extension of positive linear functionals

Abstract: Abstract. A necessary and sufficient condition to extend a continuous linear real functionals which is positive with respect to a semi-group defined on a subspace of a linear space is discussed in this paper. The case of a closed subspace of a Banach space is also discussed.

Cited by 3 publications

References 2 publications

Existence and extensions of positive linear operators

Existence and extensions of positive linear operators

$C^*$-algebras: The (Quantum) Path Less Traveled

The Sandwich Theorem for Sublinear and Superlinear Functionals

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