The Experts below are selected from a list of 291 Experts worldwide ranked by ideXlab platform
Anke Kalauch - One of the best experts on this subject based on the ideXlab platform.
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Projection bands and atoms in pervasive pre-Riesz spaces
Positivity, 2020Co-Authors: Anke Kalauch, Helena MalinowskiAbstract:In Vector Lattices, the concept of a projection band is a basic tool. We deal with projection bands in the more general setting of an Archimedean pre-Riesz space X . We relate them to projection bands in a Vector Lattice cover Y of X . If X is pervasive, then a projection band in X extends to a projection band in Y , whereas the restriction of a projection band B in Y is not a projection band in X , in general. We give conditions under which the restriction of B is a projection band in X . We introduce atoms and discrete elements in X and show that every atom is discrete. The converse implication is true, provided X is pervasive. In this setting, we link atoms in X to atoms in Y . If X contains an atom $$a>0$$ a > 0 , we show that the principal band generated by a is a projection band. Using atoms in a finite dimensional Archimedean pre-Riesz space X , we establish that X is pervasive if and only if it is a Vector Lattice.
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Vector Lattice covers of ideals and bands in pre riesz spaces
Quaestiones Mathematicae, 2019Co-Authors: Anke Kalauch, Helena MalinowskiAbstract:Pre-Riesz spaces are ordered Vector spaces which can be order densely embedded into Vector Lattices, their so-called Vector Lattice covers. Given a Vector Lattice cover Y for a pre-Riesz space X, we address the question how to find Vector Lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a Vector Lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y .Key words: Order ideal, band, pre-Riesz space, ordered Vector space, Vector Lattice cover, order dense, extension ideal, extension band, pervasive.
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bands in partially ordered Vector spaces with order unit
Positivity, 2015Co-Authors: Anke Kalauch, Bas Lemmens, Onno Van GaansAbstract:In an Archimedean directed partially ordered Vector space X, one can define the concept of a band in terms of disjointness. Bands can be studied by using a Vector Lattice cover Y of X. If X has an order unit, Y can be represented as a subspace of \(C(\Omega )\), where \(\Omega \) is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of \(\Omega \). We also analyze two methods to extend bands in X to \(C(\Omega )\) and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered Vector space with a closed generating cone, the number of bands is bounded by \(\frac{1}{4}2^{2^n}\) for \(n\ge 2\). We also construct examples of \((n+1)\)-dimensional partially ordered Vector spaces with \(\left( {\begin{array}{c}2n\\ n\end{array}}\right) +2\) bands. This shows that there are n-dimensional partially ordered Vector spaces that have more bands than an n-dimensional Archimedean Vector Lattice when \(n\ge 4\).
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bands in partially ordered Vector spaces with order unit
arXiv: Functional Analysis, 2014Co-Authors: Anke Kalauch, Bas Lemmens, Onno Van GaansAbstract:In an Archimedean directed partially ordered Vector space $X$ one can define the concept of a band in terms of disjointness. Bands can be studied by using a Vector Lattice cover $Y$ of $X$. If $X$ has an order unit, $Y$ can be represented as $C(\Omega)$, where $\Omega$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint complements, in terms of subsets of $\Omega$. We also analyze two methods to extend bands in $X$ to $C(\Omega)$ and show how the carriers of a band and its extensions are related. We use the results to show that in each $n$-dimensional partially ordered Vector space with a closed generating cone, the number of bands is bounded by $\frac{1}{4}2^{2^n}$ for $n\geq 2$. We also construct examples of $(n+1)$-dimensional partially ordered Vector spaces with ${2n\choose n}+2$ bands. This shows that there are $n$-dimensional partially ordered Vector spaces that have more bands than an $n$-dimensional Archimedean Vector Lattice when $n\geq 4$.
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Ideals and bands in pre-Riesz spaces
Positivity, 2008Co-Authors: Onno Van Gaans, Anke KalauchAbstract:In a Vector Lattice, ideals and bands are well-investigated subjects. We study similar notions in a pre-Riesz space. The pre-Riesz spaces are exactly the order dense linear subspaces of Vector Lattices. Restriction and extension properties of ideals, solvex ideals and bands are investigated. Since every Archimedean directed partially ordered Vector space is pre-Riesz, we establish properties of ideals and bands in such spaces.
Marat Pliev - One of the best experts on this subject based on the ideXlab platform.
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$C$-compact orthogonally additive operators in Vector Lattices
arXiv: Functional Analysis, 2019Co-Authors: Marat Pliev, Martin R. WeberAbstract:We consider $C$-compact orthogonally additive operators in Vector Lattices. In the first part of the article we present some examples of $C$-compact operators defined on a Vector Lattice and taking value in a Banach space. It is shown that the set of all $C$-compact orthogonally additive operators from a Vector Lattice $E$ to an order continuous Banach Lattice $F$ is a projection band in the Vector Lattice of all regular orthogonally additive operators from $E$ to $F$. In second part of the article we introduce a new class of Vector Lattices, called $C$-complete, and investigate orthogonally additive operators defined on those Lattices. We show that any laterally-to-norm continuous $C$-compact orthogonally additive operator from a $C$-complete Vector Lattice $E$ to a Banach space $X$ is narrow, which generalizes a result of Pliev and Popov. Finally we prove that the sum $T+S$ of a laterally-to-norm continuous $C$-compact operator $T\colon E\to X$ from a $C$-complete Vector Lattice $E$ to a Banach space $X$ and a narrow operator $S\colon E\to X $ is narrow.
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Atomic operators in Vector Lattices
arXiv: Functional Analysis, 2019Co-Authors: Ralph Chill, Marat PlievAbstract:In this paper we introduce a new class of operators on Vector Lattices. We say that a linear or nonlinear operator $T$ from a Vector Lattice $E$ to a Vector Lattice $F$ is atomic if there exists a Boolean homomorphism $\Phi$ from the Boolean algebra $\mathfrak{B}(E)$ of all order projections on $E$ to $\mathfrak{B}(F)$ such that $T\pi=\Phi(\pi)T$ for every order projection $\pi\in\mathfrak{B}(E)$. We show that the set of all atomic operators defined on a Vector Lattice $E$ with the principal projection property and taking values in a Dedekind complete Vector Lattice $F$, is a band in the Vector Lattice of all regular orthogonally additive operators from $E$ to $F$. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.
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Narrow and C-compact Orthogonally Additive Operators in Lattice-Normed Spaces
Results in Mathematics, 2019Co-Authors: Marat Pliev, Faruk Polat, Martin WeberAbstract:In this article we consider orthogonally additive operators on Lattice-normed spaces. In the first part of the article we present some examples of narrow, laterally-to-norm continuous and C -compact operators defined on a Lattice-normed space and taking value in a Banach space. We show that any laterally-to-norm continuous narrow orthogonally additive operator defined on a decomposable Lattice-normed space ( V , E ) over an atomic Vector Lattice E with the projection property is equal to zero. In the second part we prove that the sum of two orthogonally additive operators $$T+S$$ T + S defined on a order complete, decomposable Lattice-normed space V and taking value in Banach space X , where $$T:V\rightarrow X$$ T : V → X is a laterally-to-norm continuous C -compact operator and $$S:V\rightarrow X$$ S : V → X is a narrow operator, is a narrow operator as well.
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Order Unbounded Orthogonally Additive Operators in Vector Lattices
Mediterranean Journal of Mathematics, 2018Co-Authors: Marat Pliev, Kawtar RamdaneAbstract:In this paper, we introduce a new class of operators in Vector Lattices. We say that orthogonally additive operator T from Vector Lattice E to Vector Lattice F is laterally-to-order bounded if for any element x of E an operator T maps the set $$\mathcal {F}_{x}$$ of all fragments of x onto an order bounded subset of F. We get a Lattice calculus of orthogonally additive laterally-to-order bounded operators defined on a Vector Lattice and taking values in a Dedekind complete Vector Lattice. It turns out that these operators, in general, are not order bounded. We investigate the band of laterally continuous orthogonally additive operators and obtain formulas for the order projection onto this band. We consider the procedure of the extension of an orthogonally additive operator from a lateral ideal to the whole space. Finally we obtain conditions on the integral representability for a laterally-to-order bounded orthogonally additive operator.
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On extensions of some nonlinear maps in Vector Lattices
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Nariman Abasov, Marat PlievAbstract:Abstract We show that an Urysohn Lattice pre-homomorphism defined on a normal subLattice D of a Vector Lattice E can be extended to the whole space E and the extended operator is an Urysohn Lattice homomorphism. We introduce a new class of nonlinear operators which called φ -operators and describe some of their properties. Finally we investigate a structure of positive orthogonally additive operators dominated by an Urysohn Lattice homomorphism defined on a Vector Lattice E and taking value in Dedekind complete Vector Lattice F .
Onno Van Gaans - One of the best experts on this subject based on the ideXlab platform.
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bands in partially ordered Vector spaces with order unit
Positivity, 2015Co-Authors: Anke Kalauch, Bas Lemmens, Onno Van GaansAbstract:In an Archimedean directed partially ordered Vector space X, one can define the concept of a band in terms of disjointness. Bands can be studied by using a Vector Lattice cover Y of X. If X has an order unit, Y can be represented as a subspace of \(C(\Omega )\), where \(\Omega \) is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of \(\Omega \). We also analyze two methods to extend bands in X to \(C(\Omega )\) and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered Vector space with a closed generating cone, the number of bands is bounded by \(\frac{1}{4}2^{2^n}\) for \(n\ge 2\). We also construct examples of \((n+1)\)-dimensional partially ordered Vector spaces with \(\left( {\begin{array}{c}2n\\ n\end{array}}\right) +2\) bands. This shows that there are n-dimensional partially ordered Vector spaces that have more bands than an n-dimensional Archimedean Vector Lattice when \(n\ge 4\).
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bands in partially ordered Vector spaces with order unit
arXiv: Functional Analysis, 2014Co-Authors: Anke Kalauch, Bas Lemmens, Onno Van GaansAbstract:In an Archimedean directed partially ordered Vector space $X$ one can define the concept of a band in terms of disjointness. Bands can be studied by using a Vector Lattice cover $Y$ of $X$. If $X$ has an order unit, $Y$ can be represented as $C(\Omega)$, where $\Omega$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint complements, in terms of subsets of $\Omega$. We also analyze two methods to extend bands in $X$ to $C(\Omega)$ and show how the carriers of a band and its extensions are related. We use the results to show that in each $n$-dimensional partially ordered Vector space with a closed generating cone, the number of bands is bounded by $\frac{1}{4}2^{2^n}$ for $n\geq 2$. We also construct examples of $(n+1)$-dimensional partially ordered Vector spaces with ${2n\choose n}+2$ bands. This shows that there are $n$-dimensional partially ordered Vector spaces that have more bands than an $n$-dimensional Archimedean Vector Lattice when $n\geq 4$.
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Ideals and bands in pre-Riesz spaces
Positivity, 2008Co-Authors: Onno Van Gaans, Anke KalauchAbstract:In a Vector Lattice, ideals and bands are well-investigated subjects. We study similar notions in a pre-Riesz space. The pre-Riesz spaces are exactly the order dense linear subspaces of Vector Lattices. Restriction and extension properties of ideals, solvex ideals and bands are investigated. Since every Archimedean directed partially ordered Vector space is pre-Riesz, we establish properties of ideals and bands in such spaces.
Anna Rita Sambucini - One of the best experts on this subject based on the ideXlab platform.
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L^p spaces in Vector Lattices and applications
Mathematica Slovaca, 2017Co-Authors: Antonio Boccuto, Domenico Candeloro, Anna Rita SambuciniAbstract:$L^p$ spaces are investigated for Vector Lattice-valued functions, with respect to filter convergence. As applications, some classical inequalities are extended to the Vector Lattice context, and some properties of the Brownian Motion and the Brownian Bridge are studied, to solve some stochastic differential equations.
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filter convergence and decompositions for Vector Lattice valued measures
Mediterranean Journal of Mathematics, 2015Co-Authors: Domenico Candeloro, Anna Rita SambuciniAbstract:Filter convergence of Vector Lattice-valued measures is considered, in order to deduce theorems of convergence for their decompositions. First the σ-additive case is studied, without particular assumptions on the filter; later the finitely additive case is faced, first assuming uniform s-boundedness (without restrictions on the filter), then relaxing this condition but imposing stronger properties on the filter. In order to obtain the last results, a Schur-type convergence theorem, obtained in Boccuto et al. (Math Slovaca 62(6):1145–1166, 2012), is used.
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vitali type theorems for filter convergence related to Vector Lattice valued modulars and applications to stochastic processes
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Antonio Boccuto, Domenico Candeloro, Anna Rita SambuciniAbstract:A Vitali-type theorem for Vector Lattice-valued modulars with respect to filter convergence is proved. Some applications are given to modular convergence theorems for moment operators in the Vector Lattice setting, and also for the Brownian motion and related stochastic processes.
Helena Malinowski - One of the best experts on this subject based on the ideXlab platform.
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Projection bands and atoms in pervasive pre-Riesz spaces
Positivity, 2020Co-Authors: Anke Kalauch, Helena MalinowskiAbstract:In Vector Lattices, the concept of a projection band is a basic tool. We deal with projection bands in the more general setting of an Archimedean pre-Riesz space X . We relate them to projection bands in a Vector Lattice cover Y of X . If X is pervasive, then a projection band in X extends to a projection band in Y , whereas the restriction of a projection band B in Y is not a projection band in X , in general. We give conditions under which the restriction of B is a projection band in X . We introduce atoms and discrete elements in X and show that every atom is discrete. The converse implication is true, provided X is pervasive. In this setting, we link atoms in X to atoms in Y . If X contains an atom $$a>0$$ a > 0 , we show that the principal band generated by a is a projection band. Using atoms in a finite dimensional Archimedean pre-Riesz space X , we establish that X is pervasive if and only if it is a Vector Lattice.
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Vector Lattice covers of ideals and bands in pre riesz spaces
Quaestiones Mathematicae, 2019Co-Authors: Anke Kalauch, Helena MalinowskiAbstract:Pre-Riesz spaces are ordered Vector spaces which can be order densely embedded into Vector Lattices, their so-called Vector Lattice covers. Given a Vector Lattice cover Y for a pre-Riesz space X, we address the question how to find Vector Lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a Vector Lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y .Key words: Order ideal, band, pre-Riesz space, ordered Vector space, Vector Lattice cover, order dense, extension ideal, extension band, pervasive.
