scholarly journals Spaces generated by the cone of sublinear operators

2018 ◽  
Vol 10 (2) ◽  
pp. 376-386
Author(s):  
A. Slimane
Keyword(s):  
Convex Cone ◽  
Adjoint Operator ◽  
Analytic Form ◽  
Linear Operators ◽  
Order Continuity ◽  
Riesz Spaces ◽  
Sublinear Operators ◽  
Banach Theorem ◽  
Homogeneous Operators ◽  
Order Boundedness

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.

scholarly journals Bidual Spaces and Reflexivity of Real Normed Spaces

2014 ◽  
Vol 22 (4) ◽  
pp. 303-311
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Uniform Boundedness ◽  
Linear Operators ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Boundedness Theorem ◽  
Natural Mapping ◽  
Real Normed Space ◽  
Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

scholarly journals Boundary problems for Riccati and Lyapunov equations

1986 ◽  
Vol 29 (1) ◽  
pp. 15-21 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Boundary Condition ◽  
Boundary Problem ◽  
Transport Theory ◽  
Adjoint Operator ◽  
Linear Operators ◽  
Dimensional Case ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Image Position

The resolution problem of the systemwhere U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problemwhere t ∈[0,b].In this paper we consider the more general boundary problemwhere all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).

scholarly journals On the extension of linear operators

2001 ◽  
Vol 28 (10) ◽  
pp. 621-623 ◽  
Author(s):  
John J. Saccoman
Keyword(s):  
Extension Theorem ◽  
Linear Operators ◽  
Two Dimensional ◽  
Linear Functionals ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Banach Theorem

It is well known that the Hahn-Banach theorem, that is, the extension theorem for bounded linear functionals, is not true in general for bounded linear operators. A characterization of spaces for which it is true was published by Kakutani in 1940. We summarize Kakutani's work and we give an example which demonstrates that his characterization is not valid for two-dimensional spaces.

Riesz spaces with the order-continuity property. I

1977 ◽  
Vol 81 (1) ◽  
pp. 31-42 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Riesz Space ◽  
Continuity Property ◽  
Sequential Order ◽  
Order Continuity ◽  
Linear Map ◽  
Riesz Spaces ◽  
Quotient Spaces ◽  
Positive Linear Map ◽  
Order Continuous

A Riesz space E has the (sequential) order-continuity property if every positive linear map from E to an Archimedean Riesz space is (sequentially) order-continuous. This is the case if and only if the canonical maps from E to its Archimedean quotient spaces are all (sequentially) order-continuous. I relate these properties to others that have been described elsewhere.

Riesz spaces with the order-continuity property. II

1978 ◽  
Vol 83 (2) ◽  
pp. 211-223 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Internal Structure ◽  
Riesz Space ◽  
Continuity Property ◽  
Order Continuity ◽  
Linear Map ◽  
Riesz Spaces ◽  
Positive Linear Map ◽  
Order Continuous

I continue to investigate Riesz spaces E with the property that every positive linear map from E to an Archimedean Riesz space is sequentially order-continuous. In order to give a criterion for the product of such spaces to be another, we are forced to investigate their internal structure, and to develop an ordinal hierarchy of such spaces.

The Hahn-Banach theorem for non-Archimedean-valued fields

1952 ◽  
Vol 48 (1) ◽  
pp. 41-45 ◽  
Author(s):  
A. W. Ingleton
Keyword(s):  
Banach Spaces ◽  
Normed Linear Space ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Necessary Condition ◽  
Linear Functionals ◽  
Banach Theorem ◽  
Necessary And Sufficient ◽  
Special Case

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.

scholarly journals Extension and inversion of extended orthomorphisms on Riesz spaces

1984 ◽  
Vol 37 (2) ◽  
pp. 223-242 ◽  
Author(s):  
Michel Duhoux ◽  
Mathieu Meyer
Keyword(s):  
Riesz Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Dense Ideal ◽  
Riesz Spaces ◽  
Dense Domain ◽  
And Inversion

AbstractLet E be an Archimedean Riesz space and let Orth∞(E) be the f-algebra consisting of all extended orthomorphisms on E, that is, of all order bounded linear operators T:D→E, with D an order dense ideal in E, such that T(B∩D) ⊆ B for every band B in E. We give conditions on E and on a Riesz subspace F of E insuring that every T ∈ Orth∞(F) can be extended to some ∈ Orth∞(E), and we also consider the problem of inversing an extended orthomorphism on its support. The same problems are also studied in the case of σ-orthomorphisms, that is, extended orthomorphisms with a super order dense domain. Furthermore, some applications are given.

scholarly journals On the sublinear operators factoring throughLq

2004 ◽  
Vol 2004 (50) ◽  
pp. 2695-2704
Author(s):  
Lahcène Mezrag ◽  
Abdelmoumene Tiaiba
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Positive Constant ◽  
Bounded Linear Operator ◽  
Bounded Operator ◽  
Linear Operators ◽  
Sublinear Operator ◽  
Bounded Linear ◽  
Sublinear Operators

Let0<p≤q≤+∞. LetTbe a bounded sublinear operator from a Banach spaceXinto anLp(Ω,μ)and let∇Tbe the set of all linear operators≤T. In the present paper, we will show the following. LetCbe a positive constant. For alluin∇T,Cpq(u)≤C(i.e.,uadmits a factorization of the formX→u˜Lq(Ω,μ)→MguLq(Ω,μ), whereu˜is a bounded linear operator with‖u˜‖≤C,Mguis the bounded operator of multiplication byguwhich is inBLr+(Ω,μ)(1/p=1/q+1/r),u=Mgu∘u˜andCpq(u)is the constant ofq-convexity ofu) if and only ifTadmits the same factorization; This is under the supposition that{gu}u∈∇Tis latticially bounded. Without this condition this equivalence is not true in general.

scholarly journals Self adjoint operators and matrix measures

1971 ◽  
Vol 4 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Spectral Representation ◽  
Adjoint Operator ◽  
Multiplication Operator ◽  
Linear Operators ◽  
Positive Matrix ◽  
Adjoint Operators ◽  
Matrix Measures

Given a self adjoint operator, T, on a Hilbert space H, and given an integer n ≥ 1, we produce a collection , N ∈ L, of n × n positive matrix measures and a unitary map U: such that UTU−1, restricted to the co-ordinate space , is the multiplication operator F(t) → tF(t) in that space. This is a generalization of the spectral representation theory of Dunford and Schwartz, Linear operators, II (1966).

scholarly journals Certain types of linear operators on probabilistic Hilbert space

2015 ◽  
Vol 3 (2) ◽  
pp. 81
Author(s):  
Rana Al-Muttalibi ◽  
Radhi M.A Ali
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Real Hilbert Space ◽  
Bounded Operator ◽  
Adjoint Operator ◽  
Linear Operators ◽  
Continuous Operators

<p>The purpose of this paper is to introduce some definitions, properties and basic results that show the relation between F-bounded of linear operator in probabilistic Hilbert space and bounded operator in norm. In the paper, we prove that the adjoint operator in probabilistic Hilbert space is bounded. The notion of the continuous operators in probabilistic Hilbert space and some basic results are given. In addition, we note that every operator in probabilistic real Hilbert space is a self-adjoint Operator.</p>

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