Ограниченные ортоморфизмы между локально солидными векторными решетками

The main aim of the present note is to consider bounded orthomorphisms between locally solid vector lattices. We establish a version of the remarkable Zannen theorem regarding equivalence between orthomorphisms and the underlying vector lattice for the case of all bounded orthomomorphisms. Furthermore, we investigate topological and ordered structures for these classes of orthomorphisms, as well. In particular, we show that each class of bounded orthomorphisms possesses the Levi or the $AM$-properties if and only if so is the underlying locally solid vector lattice. Moreover, we establish a similar result for the Lebesgue property, as well.

Каждая латеральная полоса является ядром положительного ортогонально аддитивного оператора

{In this paper we continue a study of relationships between the lateral partial order $\sqsubseteq$ in a vector lattice (the relation $x \sqsubseteq y$ means that $x$ is a fragment of $y$) and the theory of orthogonally additive operators on vector lattices. It was shown in~\cite{pMPP} that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice $E$ and a lateral band $G$ of~$E$, there exists a vector lattice~$F$ and a positive, disjointness preserving orthogonally additive operator $T \colon E \to F$ such that ${\rm ker} \, T = G$. As a consequence, we partially resolve the following open problem suggested in \cite{pMPP}: Are there a vector lattice~$E$ and a lateral ideal in $E$ which is not equal to the kernel of any positive orthogonally additive operator $T\colon E\to F$ for any vector lattice $F$?

Orthogonally Biadditive Operators

In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator.

On Bilinear Narrow Operators

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×F→W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all x∈E,y∈F. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is.

Hahn-Banach-Type Theorems and Subdifferentials for Invariant and Equivariant Order Continuous Vector Lattice-Valued Operators with Applications to Optimization

We give some versions of Hahn-Banach, sandwich, duality, Moreau--Rockafellar-type theorems, optimality conditions and a formula for the subdifferential of composite functions for order continuous vector lattice-valued operators, invariant or equivariant with respect to a fixed group G of homomorphisms. As applications to optimization problems with both convex and linear constraints, we present some Farkas and Kuhn-Tucker-type results.

Convergence structures and locally solid topologies on vector lattices of operators

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) from E into F. Using this, it follows that $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) . Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.

Spanning With Binary Options: A Vector Lattice Approach

The issue of portfolio insurance is one of the prime concerns of the investors who want to insure their asset at minimum or appropriate cost. Static hedging with binary options is a popular strategy that has been explored in various option models (see e.g. (2; 3; 4; 7)). In this thesis, we propose a static hedging algorithm for discrete time models. Our algorithm is based on a vector lattice technique. In chapter 1, we give the necessary background on the theory of vector lattices and the theory of options. In chapter 2, we reveal the connection of lattice-subspaces with the minimum-cost portfolio insurance strategy. In chapter3, we outline our algorithm and give applications to binomial and trinomial option models. In chapter 4, we perform simulations and analyze the hedging errors of our algorithm for European, Barrier, Geometric Asian, Arithmetic Asian, and Lookback options. The study has revealed that static hedging could be suitable strategy for the European, Barrier, and Geometric Asian options as these options have shown less inclination to the rollover effect.

Spanning With Binary Options: A Vector Lattice Approach

The issue of portfolio insurance is one of the prime concerns of the investors who want to insure their asset at minimum or appropriate cost. Static hedging with binary options is a popular strategy that has been explored in various option models (see e.g. (2; 3; 4; 7)). In this thesis, we propose a static hedging algorithm for discrete time models. Our algorithm is based on a vector lattice technique. In chapter 1, we give the necessary background on the theory of vector lattices and the theory of options. In chapter 2, we reveal the connection of lattice-subspaces with the minimum-cost portfolio insurance strategy. In chapter3, we outline our algorithm and give applications to binomial and trinomial option models. In chapter 4, we perform simulations and analyze the hedging errors of our algorithm for European, Barrier, Geometric Asian, Arithmetic Asian, and Lookback options. The study has revealed that static hedging could be suitable strategy for the European, Barrier, and Geometric Asian options as these options have shown less inclination to the rollover effect.

Sparse representation of vectors in lattices and semigroups

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the $$\ell _0$$ ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal $$\ell _0$$ ℓ 0 -norm of solutions to systems $$A\varvec{x}=\varvec{b}$$ A x = b , where $$A\in \mathbb {Z}^{m\times n}$$ A ∈ Z m × n , $${\varvec{b}}\in \mathbb {Z}^m$$ b ∈ Z m and $$\varvec{x}$$ x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with $$\ell _0$$ ℓ 0 -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over $$\mathbb {R}$$ R , to other subdomains such as $$\mathbb {Z}$$ Z . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.