Order Complete Vector Lattice Research Articles

A representation of sup-completion

It was showed by Donner in [Extension of positive operators and Korovkin theorems, Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X X may be embedded into a cone X s X^s , called the sup-completion of X X . We show that if one represents the universal completion of X X as C ∞ ( K ) C^\infty (K) , then X s X^s is the set of all continuous functions from K K to [ − ∞ , ∞ ] [-\infty ,\infty ] that dominate some element of X X . This provides a functional representation of X s X^s , as well as an easy alternative proof of its existence.

On the diagonal of Riesz operators on Banach lattices

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators.

Diffuse orthogonally additive operators

A regular orthogonally additive operator is a diffuse operator if it is disjoint from all operators in the band generated by the disjointness preserving operators. We present a criterion for principal lateral projections in an order complete vector lattice $E$ to be disjoint. We also state a criterion for a regular orthogonally additive operator to be diffuse. A criterion for the regularity of an integral Urysohn operator acting on ideal spaces of measurable functions is also presented. This criterion is used to show that an integral operator is diffuse. Examples of vector lattices are considered in which the sets of diffuse operators consist only of the zero element. The general form of an order projection operator onto the band generated by the disjointness preserving operators is found. Bibliography: 47 titles.

On extension of abstract Urysohn operators

We consider the extension of an orthogonally additive operator from a lateral ideal and a lateral band to the whole space. We prove in particular that every orthogonally additive operator, extended from a lateral band of an order complete vector lattice, preserves lateral continuity, narrowness, compactness, and disjointness preservation. These results involve the strengthening of a recent theorem about narrow orthogonally additive operators in vector lattices.

$\varepsilon k^0$-Subdifferentia1s of Convex Functions

The paper as a contribution to convex analysis in ordered linear topological spaces. For any convex function f from a Banach space X into a partially ordered one Y endowed with a convex cone K some properties of the \varepsilon k^0 -subdifferential \partial ^\leq_{\varepsilon k^0}f(x) of f are examined. The non-emptyness of \partial ^\leq_{\varepsilon k^0}f(x) is proved, whenever Y is a normal order complete vector lattice and f belongs to the class of functions which are continuous and convex with respect to the cone K . For the real-valued case Bronsted and Rockafellar have proved that the set of subgradients of a lower semicontinuous function f on a Banach space X is dense in the set of \varepsilon -subgradients [21]. We deduce a similar result for a class of \varepsilon k^0 -subdifferentials of functions which takes values in an ordered linear topological space Y .

Nonsmooth analysis and optimization on partially ordered vector spaces

Interval‐Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz‐type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.

Saddlepoint Problems in Nondifferentiable Programming

Abstract Saddlepoint optimality conditions are derived for a class of nondifferentiable programming problems of the form min f(x) subject to x∈A, g(x)∈B, where A ⊆ X, a Banach space, and B ⊆ V, an order complete vector lattice. To establish many of the results the functions delimiting the problem are assumed to be order-Lipschitz, a property which we show is related to other Lipschitz-type definitions in the literature. A nonsmooth analysis for order-Lipschitz functions is included; in particular, directional derivatives, generalized gradients, optimality conditions and results concerning a calculus of generalized gradients are discussed. The saddlepoint optimality conditions are shown to compare favorably with other optimality conditions in the literature.

Generalized differentiability for a class of nondifferentiable operators with applications to nonsmooth optimization

AbstractA theory of generalized gradients is presented for a class of Lipschitz vector-valued mappings from a Banach space to a locally convex order complete vector lattice. Necessary optimality conditions are obtained for nonconvex programming problems on Banach spaces with vector- valued operator constraints and/or an arbitrary set constraint. Sufficient optimality conditions are also obtained under mild convexity assumptions.

Nonsmooth analysis of vector-valued mappings with contributions to nondifferentiable programming

We define order Lipschitz mappings from a Banach space to an order complete vector lattice and present a nonsmooth analysis for such functions. In particular, we establish properties of a generalized directional derivative and gradient and derive results concerning a calculus of generalized gradients (i.e., calculation of the generalized gradient of f when f = f1 + f2, f = f · 2, etc.). We show the relevance of the above analysis to nondifferentiaile programming by deriving optimality conditions for problems of the form min f(x) subject to x [euro] S. For S arbitrary we state the results in terms of cones of displacement of the feasible region at the optimal point; when S ={x ∊ A|g(x) ∊ B}, we obtain Kuhn-Tucker type results.

On the Radon-Nikodym theorem for measures with values in vector lattices

Measures with values in a countably order-complete vector lattice are considered. The underlying σ-algebra is assumed to be σ-isomorphic to the Borel sets of the real line. Given one such measure, densities are searched which are not necessarily scalar-valued for smaller measures. The results can be used to prove the existence of a least upper bound for two such measures.

Factorization by lattice homomorphisms

Wolfgang Arendt Mathematisches Institut der UnJversit~t Ttibingen, Auf der Morgenstelle 10, D-7400 TiJbingen 1, Federal Republic of Germany The purpose of the present note is to prove the following: Let V be a lattice homomorphism from an order complete vector lattice E into E. Then given a positive linear mapping S on E every positive linear mapping T dominated by

Zum satz von ljapunov in vektorverbänden

Let K be a closed convex order-bounded set of an order-complete vector lattice and let A be a him continuous linear operator. Then the equality AK = A(ext K) is proved, where ext K is the set of all extremal points of K. It is shown that various generalizations of the Ljapunov-theorem on the range of vector-measures are special cases of this general statement.

The Jordan Decomposition of Vector-Valued Measures

This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and $\mathcal {A}$ is a Boolean algebra. Then an additive set function $\mu :\mathcal {A} \to L$ can be expressed as the difference of two positive additive measures if and only if $\mu (\mathcal {A})$ is order bounded. A sufficient condition for a countably additive set function with values in ${c_0}(\Gamma )$, for any set $\Gamma$, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set.

The Jordan decomposition of vector-valued measures

This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and 6 is a Boolean algebra. Then an additive set function ,u: 6 -* L can be expressed as the difference of two positive additive measures if and only if ,u(d) is order bounded. A sufficient condition for a countably additive set function with values in c0(F), for any set F, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set. We are concerned here with vector-valued additive set functions defined on some sort of Boolean algebra with vector-values in a (generally order complete) vector lattice. The purpose of this note is to expose conditions that insure such measures can be written as a difference of positive measures, i.e., conditions for a vector-valued Jordan decomposition theorem to hold. For this reason, a measure that can be expressed as the difference of two positive additive measures will be called decomposable. The decomposability of vector measures per se was first studied by C. E. Rickart in a 1943 Duke Mathematical Journal article where he established a Lebesgue decomposition theorem for the class of strongly bounded additive measures. This result was later re-established (although it was not realized at the time) by J. J. Uhl, Jr. [10] who also presented a Yosida-Hewitt decomposition theorem for strongly bounded measures. In [3], Diestel and Faires exhibited several decomposition theorems of the Jordan type, however, they did not give necessary and sufficient conditions for the decomposability of a vector measure. Our first result supplies these conditions, although, as is indicated, only one part of the proof is new. So, let E be a Boolean algebra, S. the vector lattice of simple functions over E endowed with the uniform norm, and L a vector lattice. If tt: C -> L is an additive set function, then the operator T : S, -> L associated with yt is defined by n n T(, aic) = ai fL(ai) Received by the editors January 24, 1975 and, in revised form, February 12, 1976. AMS (MOS) subject classifications (1970). Primary 46G10; Secondary 28A25. Copyright 6) 1977, American Mathematical Society

A duality theorem for a convex programming problem in order complete vector lattices

The problem “minimize f(x) -g(x)” is studied, where f is a convex function and g a concave function of a real vector space X into an order complete vector lattice Y. We define Y-valued functions f”(T) and gc( T) on the real vector space of linear mappings of X into Y and associate with the minimization problem the maximization problem “maximize gC(T) fC(T).” For X = R” and Y = R Fenchel [2, 31 showed that these programs are dual to each other. Our main result will be that, iff g is bounded below, the maximization problem has an optimal solution and the extreme values of both objective functions are equal. As a direct consequence we get a condition, saying that a point x is an optimal solution of our minimization problem if and only if f and g have a common subgradient at x. This result extends a theorem by Valadier [9], concerning the subdifferentiability of a convex function with values in an order complete vector lattice. Furthermore, we apply our theorem to prove an extension of the KuhnTucker Theorem for order complete vector lattices. To this we generalize an idea from Rockafellar [7]. Our result is similar to one given by Ritter [6]. An example is given, showing that the hypothesis of order completeness is necessary for our theorems.

Konjugierte operatoren und subdifferentiale

In this paper we give separation theorems for convex sets of a product space. Separation is carried out by linear operators. With these theorems we prove several assertions on conjugate operators and subdifferentials of operators (which map a vector space into an order complete vector lattice), where we use the definition of conjugate operators as done by Zowe. Simultaneously we generalize some of his results to such operators. Moreover we prove that an order complete vector lattice is a vector lattice with certain separation properties.