arXiv subjects

M. A. A. Marabeh

Publications and source records attributed to M. A. A. Marabeh.

6 recordsLinked to original sources

arXiv2017

Unbounded $p$-convergence in Lattice-Normed Vector Lattices

A net $x_α$ in a lattice-normed vector lattice $(X,p,E)$ is unbounded $p$-convergent to $x\in X$ if $p(|x_α-x|\wedge u)\xrightarrow{o} 0$ for every $u\in X_+$. This convergence has been investigated recently for $(X,p,E)=(X,\lvert\cdot \rvert,X)$ under the name of $uo$-convergence, for $(X,p,E)=(X,\lVert\cdot\rVert,{\mathbb R})$ under the name of $un$-convergence, and also for $(X,p,{\mathbb R}^{X^*})$, where $p(x)[f]:=|f|(|x|)$, under the name $uaw$-convergence. In this paper we study general properties of the unbounded $p$-convergence.

math.FA↗

arXiv2017

$uτ$-Convergence in locally solid vector lattices

Let $x_α$ be a net in a locally solid vector lattice $(X,τ)$; we say that $x_α$ is unbounded $τ$-convergent to a vector $x\in X$ if $\lvert x_α-x \rvert\wedge w \xrightarrowτ 0$ for all $w\in X_+$. In this paper, we study general properties of unbounded $τ$-convergence (shortly, $uτ$-convergence). $uτ$-Convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. Besides, we introduce $uτ$-topology and study briefly metrizabililty and completeness of this topology.

math.FA↗

arXiv2017

$um$-Topology in multi-normed vector lattices

Let $\mathcal{M}=\{m_λ\}_{λ\inΛ}$ be a separating family of lattice seminorms on a vector lattice $X$, then $(X,\mathcal{M})$ is called a multi-normed vector lattice (or MNVL). We write $x_α\xrightarrow{\mathrm{m}} x$ if $m_λ(x_α-x)\to 0$ for all $λ\inΛ$. A net $x_α$ in an MNVL $X=(X,\mathcal{M})$ is said to be unbounded $m$-convergent (or $um$-convergent) to $x$ if $\lvert x_α-x \rvert\wedge u \xrightarrow{\mathrm{m}} 0$ for all $u\in X_+$. $um$-Convergence generalizes $un$-convergence \cite{DOT,KMT} and $uaw$-convergence \cite{Zab}, and specializes $up$-convergence \cite{AEEM1} and $uτ$-convergence \cite{DEM2}. $um$-Convergence is always topological, whose corresponding topology is called unbounded $m$-topology (or $um$-topology). We show that, for an $m$-complete metrizable MNVL $(X,\mathcal{M})$, the $um$-topology is metrizable iff $X$ has a countable topological orthogonal system. In terms of $um$-completeness, we present a characterization of MNVLs possessing both Lebesgue's and Levi's properties. Then, we characterize MNVLs possessing simultaneously the $σ$-Lebesgue and $σ$-Levi properties in terms of sequential $um$-completeness. Finally, we prove that any $m$-bounded and $um$-closed set is $um$-compact iff the space is atomic and has Lebesgue's and Levi's properties.

math.FA↗

arXiv2017

Order convergence in infinite-dimensional vector lattices is not topological

In this note, we show that the order convergence in a vector lattice $X$ is not topological unless $\dim X<\infty$. Furthermore, we show that, in atomic order continuous Banach lattices, the order convergence is topological on order intervals.

math.FA↗

arXiv2017

Compact-Like Operators in Lattice-Normed Spaces

A linear operator $T$ between two lattice-normed spaces is said to be $p$-compact if, for any $p$-bounded net $x_α$, the net $Tx_α$ has a $p$-convergent subnet. $p$-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, $AM$-compact operators, etc. Similar to $M$-weakly and $L$-weakly compact operators, we define $p$-$M$-weakly and $p$-$L$-weakly compact operators and study some of their properties. We also study $up$-continuous and $up$-compact operators between lattice-normed vector lattices.

math.FA↗

arXiv2017

Unbounded Norm Topology in Banach Lattices

A net $(x_α)$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\bigl\lVert\lvert x_α-x\rvert\wedge u\bigr\rVert\to 0$ for every $u\in X_+$. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff $X$ has a strong unit. Un-topology is metrizable iff $X$ has a quasi-interior point. Suppose that $X$ is order continuous, then un-topology is locally convex iff $X$ is atomic. An order continuous Banach lattice $X$ is a KB-space iff its closed unit ball $B_X$ is un-complete. For a Banach lattice $X$, $B_X$ is un-compact iff $X$ is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.

math.FA↗