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Eduard Emelyanov

Publications and source records attributed to Eduard Emelyanov.

At least 19 recordsLinked to original sources
arXiv2026

Relative uniform convergence and Archimedean property in pre-ordered vector spaces and Abelian groups

It is proved that, for a pre-ordered vector space $X$, the quotient space $(X/A,[W])$ is an Archimedeanization of $X$, where $W$ is the closure of positive wedge $X_+$ in ru-topology, $A=W\cap-W$, and $[W]$ is the quotient set of $W$ in $X/A$. A construction of an almost Archimedeanization of a pre-ordered vector space is given. Analogous results are presented for pre-ordered groups. We present an explicit example of a family of vector lattices for which a chain of spaces of infinitely small elements stabilizes first at any chosen ordinal $α\leqslantω_1$.

math.FA↗
arXiv2026

On automatic continuity of operators from ordered to topological vector spaces

We study continuity and boundedness of order-to-topology bounded and order-to topology continuous operators from ordered to topological vector spaces. Several results on automatic continuity of operators from ordered Frechet spaces to topological vector spaces are included. Levi and Lebesgue operators especially are investigated.

math.FA↗
arXiv2025

On Levi operators between normed and vector lattices

The notion of a Levi operator is an operator abstraction of the Levy property of a norm or, more generally of the Levi topology on a locally solid vector lattice. Various aspects of Levi operators have been studied recently by several authors. The present paper is devoted to Levi operators from a normed lattice to a vector lattice. It is proved that every finite rank operator is a Levi operator. An example is given showing that the sum of a positive rank one operator and a positive compact Levi operator need not to be a Levi operator. We prove that every quasi Levi operator is continuous. It is shown that the set of Levi operators on the space of convergent sequences is not complete in the operator norm. Several results concerning the domination problem for Levi operators and the relations between Levi operators and KB-spaces are established.

math.FA↗
arXiv2025

Free uniformly complete vector lattices

We define a free uniformly complete vector lattice over a set of generators and give its concrete representation as a space of continuous positively homogeneous functions.

math.FA↗
arXiv2025

Collective order convergence and collectively qualified sets of operators

Collective versions of order convergences and corresponding types of collectively qualified sets of operators in vector lattices are investigated. It is proved that collectively order to norm bounded sets are bounded in the operator norm and collectively order continuous sets are collectively order bounded.

math.FA↗
arXiv2025

Collective order boundedness of sets of operators between ordered vector spaces

It is proved that: each collectively order continuous set of operators from an Archimedean OVS with a generating cone to an OVS is collectively order bounded; and each collectively order to norm bounded set of operators from an ordered Banach space with a closed generating cone to a normed space is norm bounded. Several applications to commutative operator semigroups on ordered vector spaces are given.

math.FA↗
arXiv2024

On the extension of one-parameter operator semigroups to completions of Archimedean vector lattices

Extensions of one-parameter operator semigroups on Archimedean vector lattices to their order/ru-completions are studied. Existence and uniqueness of the extension to the ru-completion is established in the class of positive semigroups. An extension theorem for positive ru-continuos semigroups on vector lattices with property (R) is proved. This theorem allows one to abandon the requirement of ru-completeness in various results concerning positive ru-continuous semigroups.

math.FA↗
arXiv2024

On collectively L-weakly compact sets of operators

A set of bounded linear operators from a Banach space to a Banach lattice is collectively L-weakly compact whenever union of images of the unit ball is L-weakly compact. We extend the Meyer-Nieberg duality theorem to collectively L-weakly compact sets of operators, study relations between these sets and collectively almost limited sets, and discuss the domination problem for collectively compact sets and collectively L-weakly compact sets.

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arXiv2024

d-Operators in Banach Lattices

We study operators carrying disjoint bounded subsets of a Banach lattice into compact, weakly compact, and limited subsets of a Banach space. Surprisingly, these operators behave differently with classical compact, weakly compact, and limited operators. We introduce and investigate d-Andrews operators which carry disjoint bounded sets onto Dunford--Pettis sets.

math.FA↗
arXiv2024

On collectively $σ$-Levi sets of operators

A collectively $σ$-Levi set of operators is a generalization of the $σ$-Levi operator. By use of collective order convergence, we investigate relations between collectively $σ$-Levi and collectively compact sets of operators.

math.FA↗
arXiv2023

On KB and Levi operators in Banach lattices

We prove that an order continuous Banach lattice E is a KB-space if and only if each positive compact operator on E is a KB operator. We give conditions on quasi-KB (resp., quasi-Levi) operators to be KB (resp., Levi), study norm completeness and domination for these operators, and show that neither KB nor Levi operators are stable under rank one perturbations.

math.FA↗
arXiv2023

Limitedly L-weakly compact operators

We introduce new class of limitedly L-weakly compact operators from a Banach space to a Banach lattice. This class is a proper subclass of the Bourgain-Diestel operators and it contains properly the class of L-weakly compact operators. We give its efficient characterization in term of sequences, investigate the domination problem, and study the completeness of this class of operators.

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