CHOQUET BOUNDARIES IN<i>K</i>-SPACES

Kutateladze, S. S.

doi:10.1070/rm1975v030n04abeh001514

Cited by 9 publications

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“…The above theorem was stated and proved in this form in [6]. Obviously, the Riesz space in this theorem may be viewed over an arbitrary dense subfield of the reals R. We note in passing that the results to follow are preserved in the general modules admitting convex analysis (cp.…”

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confidence: 84%

On differences of Riesz homomorphisms

Kutateladze

2005

Sib Math J

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An order bounded functional on a Riesz space is a difference of Riesz homomorphisms if and only if the kernel of this functional is a Riesz subspace of the ambient Riesz space. An operator version of this fact is given. This note answers the following question: Which close hyperplane in a Banach lattice is a Riesz subspace; i.e., a subspace closed under finite meets and joins? It turns out that each hyperspace with this property is exactly the kernel of the difference of some Riesz homomorphisms on the initial Riesz space. The starting point of the present article is the celebrated Stone Theorem about the structure of Riesz subspaces in the Banach lattice C(Q, R) of continuous real functions on a compact space Q (for instance, cp. [1, Theorem 10.8.10]). This theorem may be rephrased in the above terms as follows:Stone Theorem. Each closed Riesz subspace of C(Q, R) is the intersection of the kernels of some differences of Riesz homomorphisms on C(Q, R).In view of this theorem we find it reasonable to refer to a difference of Riesz homomorphisms on a Riesz space X as a two-point relation on X.We are not obliged to assume here that the Riesz homomorphisms under study act into the reals R. The theory of dominated operators includes the classical Meyer Theorem (see [2, § 3.3.1 (5)]) which reads as follows: Each order bounded disjointness preserving operator between Riesz spaces is a twopoint relation. An analogous theorem in a slightly more general setting was established by Abramovich, Arenson, and Kitover (cp. [3, § 5.4.1]).In what follows we will consider the general Riesz homomorphisms from X to an arbitrary Kantorovich (= Dedekind complete Riesz) space Y with base a complete Boolean algebra B.So, let T : X → Y be an order bounded operator whose positive and negative parts are Riesz homomorphisms. Observe that for every band projection b ∈ B the operator bT , called a stratum of T , is a two-point relation on X and so the kernel of bT is a Riesz subspace of X. In fact, the converse is valid too. In other words, we have the following Main Theorem. An order bounded operator from a Riesz space to a Kantorovich space is a twopoint relation if and only if the kernel of its every stratum is a Riesz subspace of the ambient Riesz space.The analysis below is rather transparent if we apply a well-developed technique of "nonstandard scalarization." This technique implements the Kantorovich heuristic principle and reduces operator problems to the case of functionals. To save room, using the facts of Riesz space theory and Boolean valued analysis we will presume the terminology and notation of [4].The sketch of the proof is as follows: Using the functors of canonical embedding and ascent to the Boolean valued universe V (B) , we reduce the matter to characterizing scalar two-point relations on Riesz spaces over dense subrings of the reals R. To solve the resulting scalar problem, we use one of the formulas of subdifferential calculus [5] which is known as Decomposition Theorem. Assume that H 1 , . . . , H N are cones in a Rie...

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confidence: 84%

On differences of Riesz homomorphisms

Kutateladze

2005

Sib Math J

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“…However, μ(x) R N μ(y). In fact, applications require the following more detailed version of majorization (see [28]): 2.9. Decomposition Theorem.…”

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confidence: 99%

Multiobjective problems of convex geometry

Kutateladze

2009

Sib Math J

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Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body x, we try to maximize the volume of x and minimize the width of x simultaneously. These problems are addressed along the lines of multiple criteria decision making. We describe the Pareto-optimal solutions of isoperimetric-type vector optimization problems on using the techniques of the space of convex sets, linear majorization, and mixed volumes.

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“…Sublinear operators with values in a boundedly complete vector lattice (ΑΓ-space) were used by Kantorovich, Vulikh, and Pinsker [21] in the study of extension problems for operators, in the construction of generalized Banach limits, and for other purposes; we remark that an abstract norm [21] (with values in a ίΓ-space) is an example of a sublinear operator. Special sublinear operators are used in the study of convergence problems for sequences of positive operators [29], in the operator theory of Choquet [26], and in other problems.…”

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confidence: 99%

Sublinear Operators and Their Applications

Rubinov¹

1977

Russ. Math. Surv.

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The Hann-Banach-Kantorovich and Mazur-Orlicz theorems. IfP-X -*• Υ U {+ oo} is a subadditive positive-homogeneous operator and domP = X, then we say that Ρ is defined everywhere, or that Ρ acts from X into Y, and we write P: X -*• Y. In this subsection it is assumed that Υ is a ^Γ-space. In the study of an operator Ρ: Χ -» Υ a fundamental role is played by the Hahn-Banach-Kantorovich theorem [21], which asserts that every linear operator A o : X o -»• Y, where X o is a subspace of X, that is majorized on X o by Ρ can be extended to a linear operator 4: X -> Υ with preservation of the majorization. Kutateladze has proposed to state this theorem as follows.HAHN-BANACH-KANTOROVICH THEOREM. Let X be a vector space, X o a subspace of it, and Υ a K-space. Further, let P: X -> Υ be a subadditive positive-homogeneous operator, and P x -Ρ + δ γ (Χ 0 ) the restriction of Ρ to X o . Then dP Xo = dP + d8 Y (X o j. It is known that if a partially ordered space Υ is such that for any X and Ρ the Hahn-Banach-Kantorovich theorem holds, then Υ is a K-spa.ce.It follows directly from the Hahn-Banach-Kantorovich theorem that for any χ G X the subdifferential dP(x) of an operator Ρ at a point χ is not empty (it suffices to set X o = (λχ) λεΛ , ^4 0 (λχ) = XPx in the theorem). From this it follows that bP is not empty and that Px = sup {Ax: A £ dP). Thus, every subadditive positive-homogeneous operator P: X -> Y, where Fis a K-spacQ, is sublinear.It is not hard to show that dP(x) = dP x , where the directional derivative P' x of Ρ at χ is defined byREMARK. Fel'dman [62] has shown that dP Xo = dP + 95 ^(^ο) also when Υ has the property of chain completeness and X o is a onedimensional subspace of X. From this it follows that bP(x) Φ φ for every subadditive positive-homogeneous operator P: X ->· Y, where Υ has the property of chain completeness, and, in particular, every such operator is sublinear.An important consequence of the Hahn-Banach-Kantorovich theorem is the following proposition, due to Levin [32]. PROPOSITION 1. Let X and Ζ be vector spaces, Υ a K-space, A G X{Z, X) and P: X -> Υ a sublinear operator. Then d(P ° A) = dP ο A.(If a subadditive positive-homogeneous operator Ρ: Χ -> Υ U { + °°} is such that dom Ρ Φ Χ, then the question whether dP(x) is non-empty at a point χ (and, in particular, whether dP= dP(0) if Ρ(0) Φ + °°) is substantially more complicated. This problem was studied by Fel'dman in [62]. In investigating

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CHOQUET BOUNDARIES INK-SPACES

Cited by 9 publications

References 58 publications

On differences of Riesz homomorphisms

On differences of Riesz homomorphisms

Multiobjective problems of convex geometry

Sublinear Operators and Their Applications

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