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A continuous operator $T$ between two Banach lattices $E$ and $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we show…
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…
In this paper, we study some geometric properties in Banach lattices and the class of almost limited completely continuous operators. For example, we study Banach lattices with the property (d) and we give a new characterization of this…
Let $T$ be a bounded linear operator on a (real or complex) Banach space $X$. If $(a_n)$ is a sequence of non-negative numbers tending to 0. Then, the set of $x \in X$ such that $\|T^nx\| \geqslant a_n \|T^n\|$ for infinitely many $n$'s has…
The aim of this article is to extend results of Maslyuchenko O., Mykhaylyuk V. Popov M. 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…
In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star,…
We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by "componental operators" acting on the Hilbert spaces that form the product space. We attribute operatorial properties…
In this paper we study the Y-convexity, a property which is obtained by considering a real Banach sequence lattice Y instead of $\ell^p$ for a linear operator $T : E \rightarrow X$, where E is a Banach space and X is a Banach lattice. We…
For an operator-differential equation of the form $y^{(m)}(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the the field of $p$-adic numbers, the necessary and sufficient conditions on initial data for the Cauchy…
We introduce an ordinal index which characterizes weak compactness of operators between Banach spaces. We study when classes consisting of operators having bounded index form a closed ideal, the distinctness of the classes, and the…
We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions,…
We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all…
We investigate possible quantifications of strictly singular operators, $l_{p}$-strictly singular operators, $c_{0}$-strictly singular operators, strictly cosingular operators, $l_{p}$-strictly cosingular operators. We prove quantitative,…
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.…
In the context of general Banach spaces characterizations for the maximal monotonicity of operators with non-empty domain interior as well as stronger continuity properties for such operators are provided.
In this paper, we study the hypercyclic composition operators on weighted Banach spaces of functions defined on discrete metric spaces. We show that the only such composition operators act on the "little" spaces. We characterize the bounded…
Problem for the first order differential equation with an unbounded operator coefficient in Banach space and integral nonlocal condition is considered. An exponentially convergent algorithm is proposed and justified for the numerical…
Several recent papers investigated unbounded convergences in Banach lattices. Combine all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly…
Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (…