Marat Amurhanovich Pliev

Southern Mathematical Institute of Russian Academy of Science · Department of Functional Analysis

We investigate order projections onto different bands in the space of all regular orthogonally additive operators. In particular, we obtain formulas for calculation of the order projections onto the band generated by a directed set of positive orthogonally additive operators and onto the band of all laterally continuous operators.

Let H \mathcal {H} be a separable Hilbert space and let B ( H ) B(\mathcal {H}) be the ∗ * -algebra of all bounded linear operators on H \mathcal {H} . In the present paper, we prove that a positive/regular operator from L 1 ( 0 , 1 ) L_1(0,1) into an arbitrary separable operator ideal in B ( H ) B(\mathcal {H}) is necessarily Dunford–Pettis, exten...

We fully characterize the class of von Neumann algebras M (on a separable Hilbert space) such that any non-Dunford–Pettis operator on the predual M⁎ fixes a copy of ⊕ℓ1(ℓ2), which extends results due to Bourgain [9] and Rosenthal [49]. We also fully characterize those noncommutative symmetric spaces affiliated with different types of von Neumann al...

Let $(\mathcal{F}_n)_{n\ge 0}$ be the standard dyadic filtration on $[0,1]$. Let $\mathbb{E}_{\mathcal{F}_n}$ be the conditional expectation from $ L_1=L_1[0,1]$ onto $\mathcal{F} _n$, $n\ge 0$, and let $\mathbb{E}_{\mathcal{F} _{-1}} =0$. We present the sharp estimate for the distribution function of the martingale transform $T$ defined by \begin{...

In this article, we investigate orthogonally additive (nonlinear) operators on C-complete vector lattices which strongly includes all Dedekind complete vector lattices. In the first part of the paper, we present basic examples of orthogonally additive operators on function spaces. Then we show that an orthogonally additive map defined on a lateral...

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 biadd...

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 K...

We show that every ℓ 2 \ell _2 -strictly singular operator on the predual of any atomless hyperfinite finite von Neumann algebra M \mathcal {M} is Dunford–Pettis, which extends a Rosenthal’s theorem for the case of commutative algebra M = L ∞ ( 0 , 1 ) \mathcal {M}=L_\infty (0,1) . We also apply our result to the study of noncommutative symmetric s...

The paper contains a systematic study of the lateral partial order \(\sqsubseteq \) in a Riesz space (the relation \(x \sqsubseteq y\) means that x is a fragment of y) with applications to nonlinear analysis of Riesz spaces. We introduce and study lateral fields, lateral ideals, lateral bands and consistent subsets and show the importance of these...

In this article we consider some classes of orthogonally additive operators in Köthe–Bochner spaces in the setting of the theory of lattice-normed spaces and dominated operators. The our first main result asserts that the C-compactness of a dominated orthogonally additive operator \(S:E(X)\rightarrow F(Y)\) implies the C-compactness of its exact do...

We extend the Kalton-Rosenthal representation theorem for operators on L1(μ) to the setting of dominated operators on lattice-normed spaces. In the special case of Köthe-Bochner spaces of measurable vector-valued functions our main result asserts that every dominated operator T:E(X)→F from a Köthe-Bochner space E(X) to an order continuous Banach fu...

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...

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 Ba...

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 $T\pi=\Phi(\pi)T$ f...

We show that for any two elements x, y of a Hilbert A-module M over a locally C*-algebra A the generalized triangle equality ∣x + y∣ = ∣x∣ + ∣y∣ holds if and only if 〈x, y〉 = ∣x∣∣y∣.

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 additi...

We consider completely positive maps defined on locally C*-algebra and taking values in the space of sesquilinear forms on Hilbert C*-module M. We construct the Stinespring type representation for this type of maps and show that any two minimal Stinespring representations are unitarily equivalent.

In this paper, we introduce a new class of operators in lattice- normed spaces. We say that an orthogonally additive operator T from a lattice-normed space (V;E) to a lattice-normed space (W; F) is dominated, if there exists a positive orthogonally additive operator S from E to F such that |Tx| ≤ S |x| for any element x of (V,E). We show that under...

In this article, we investigate disjointness-preserving orthogonally additive operators in the setting of vector lattices. First, we present a formula for the band projection onto the band generated by a single positive, disjointness-preserving, order-bounded, orthogonally additive operator. Then we prove a Radon-Nikodým theorem for a positive, dis...

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...

We prove that for a Köthe–Banach space E with an order continuous norm over a finite atomless measure space (Ω,Σ,μ) and for Banach spaces X, Y, the classes of narrow and weakly functionally narrow operators from a Köthe–Bochner space E(X) to a Banach space Y are coincident. We also obtain that in the general case, without the assumption of order co...

Abstract. We study the collection of finite elements Φ 1 (U(E, F )) in the vector lattice U (E, F ) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F , where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in φ ∈ U (E, R) there...

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A-modules over locally C*-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon-Nikodym theorem for th...

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...

The "Up-and-down" theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result of the operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result we appl...

We continue the investigation of the space of dominated Urysohn operators between lattice-normed spaces which started in [2, 12]. We prove the Yosida-Hewitt type theorem for this class of operators. We also show that dominated Urysohn operator T preserve disjointness if and only if preserve disjointness its exact dominant |T|.

Projections onto several special subsets in the Dedekind complete vector lattice of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices $E$ and $F$ are considered and some new formulas are provided.

We prove an analogue of the Radon--Nikodym type theorem for $n$-tuples of the completely positive maps on Hilbert $C^*$-modules. Our results are generalization of some results of the paper M. Joi\c{t}a [J. Math. Anal. Appl. 393 (2012), 644--650].

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi:10. 1007/ s11117-016-0401-9, 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous...

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 res...

We use a new technique of measures on Boolean algebras to investigate narrow operators on vector lattices. First we prove that, under mild assumptions, every finite rank operator is strictly narrow (before it was known that such operators are narrow). Then we show that every order continuous operator from an atomless vector lattice to a purely atom...

We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order-continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic latt...

An analytical representation of order continuous functionals on a space of measurable sections of a liftable bundle of Banach spaces is given.

We continue the investigation of abstract Uryson operators in vector lattices. Using the recently proved Up-and-down theorem for order bounded, orthogonally additive operators, we consider the domination problem for AM-compact abstract Uryson operators. We obtain the Dodds-Fremlin type theorem and prove that for an AM- compact positive abstract Ury...

The aim of this article is to extend results of M.~Popov and second named author about orthogonally additive narrow operators on vector lattices. The main object of our investigations are an orthogonally additive narrow operators between lattice-normed spaces. We prove that every $C$-compact laterally-to-norm continuous orthogonally additive operat...

We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic latt...

We show that every dominated linear operator from an Banach-Kantorovich space over atomless Dedekind complete vector lattice to a sequence Banach lattice $l_p({\Gamma})$ or $c_0({\Gamma})$ is narrow. As a conse- quence, we obtain that an atomless Banach lattice cannot have a finite dimensional decomposition of a certain kind. Finally we show that i...

We show that every dominated linear operator from a Banach-Kantorovich space over atomless Dedekind complete vector lattice to a sequence Banach lattice ℓp(Γ) or c0(Γ) is narrow. As a consequence, we obtain that an atomless Banach lattice cannot have a finite-dimensional decomposition of a certain kind. Finally, we show that if a linear dominated o...

We prove a Radon-Nikodym type theorem for completely positive maps, (u; u')- covariant with respect to the dynamical system (G; η;X) on Hilbert C*-modules.

We continue the investigation of dominated Uryson operators in lattice-normed spaces started in [16]. The main subjects are laterally continuous, completely ad- ditive and C-compact dominated Uryson operators. We prove that a dominated Uryson operator is laterally continuous (completely additive) if and only if so is its exact dominant. We also pro...

We continue to investigation the space of abstract Uryson operators U(E;F), acting between vector lattices E and F.We introduce a new class of orthogonally additive, disjointness preserving operators which called Uryson lattice homomorphisms.We consider some examples of this operators and prove the Meyer type theorem. © 2015 Nariman Abasov and Mara...

The "Up-and-down" theorem which describe the structure of the Boolean alge- bra fragments of a positive orthogonally additive operator was recently proved in [14]. This result we apply to prove the decomposability of the lattice valued norm of the space DU(V,W) of all dominated Uryson operators. We obtain that, for a lattice-normed space V and a Ba...

Here we prove an analog of the Stinespring’s theorem for n-tuples of completely positive maps in Hilbert C ⋆-modules.

We provide a construction in a general vector lattice similar to martingales, however not using both integrability and norms of elements. The idea is to replace the mean value with the infimum value in the conditional expectation of a positive element. On the one hand, the construction is quite general. But on the other hand, the corresponding oper...

The aim of this note is to introduce the space DU(V,W) of dominated orthogonally additive operators on lattice-normed spaces. We prove that under some mild conditions, a dominated Uryson operator has an exact dominant and obtain formulas for calculating the exact dominant of a dominated Uryson operator.

The aim of this note is to introduce the space D_{U}(V,W) of the dominated Uryson operators on lattice-normed spaces. We prove an "Up-and-down" type theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result we apply to prove the decomposability of the lattice val...

We consider the space of abstract Uryson operators firstly introduced in [9]. We obtain the formulas for band projections on the band generated by increasing set of a positive Uryson operators and on the band generated one-dimensional abstract Uryson operators. We also calculate the laterally continuous part of a abstract Uryson operator.

We extend the notion of narrow operators to nonlinear maps on vector lattices. The main objects are orthogonally additive operators and, in particular, abstract Uryson operators. Most of the results extend known theorems obtained by O. Maslyuchenko, V. Mykhaylyuk and the second named author published in Positivity 13 (2009), pp. 459--495, for linea...

We consider the space of abstract Uryson operators in vector lattices. We calculate the laterally continuous part of a positive abstract Uryson operator.

We prove an analog of the Stinespring theorem for Hilbert modules over local C*-algebras.

The aim of this article is to extend the results of Asadi M.B, B.V.R. Bhat, G. Ramesh, K. Sumesh about completely positive maps on Hilbert C*-modules. We prove a Stinespring type theorem for a finite family of completely positive maps on Hilbert C*-modules. We also show that any two minimal Stinespring representations are unitarily equivalent.

We consider the measurable section spaces E(χ), where χ is a measurable bundle of Banach spaces and E is Köthe function spaces. We show that E(χ) space is uniformly (strict) convex if the space E and almost every all fibers χt are uniformly (strict) convex. If E(χ) is uniformly (strict) convex so is a E, moreover if a fiber χt have a density r(χt)...

The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator fro...

We show that a ring homomorphism from a local σ-C*-algebra to a local C*-algebra is a continuous mapping. Keywords and phraseslocal C*-algebras–homomorphisms–positive operators