arXiv subjects
Publications and source records attributed to Omid Zabeti.
Let $E$ and $F$ be Banach lattices. In this short note, we consider several conditions under them, the Fremlin projective tensor product $E\widehat{\otimes}F$ respects either the unbounded norm convergence or unbounded absolute weak convergence.
We prove that the Fremlin projective tensor product of Banach lattices preserves unbounded norm convergence and unbounded absolute weak convergence of elementary tensors without any additional assumptions. The main tool is a tensor-lattice estimate controlling expressions of the form \[ \lvert x_α\otimes y_β-x\otimes y\rvert \wedge w, \] where $w$ is positive in the tensor product. This provides a direct and simplified proof of permanence results previously obtained under extra hypotheses.
For an arbitrary topological space $X$, assume that $S(X)$ is the vector lattice of all equivalence classes of real-valued continuous functions on open dense subsets of $X$; it is a laterally complete vector lattice but not a normed lattice, certainly. Nevertheless, we can have the extended unbounded norm topology ($un$-topology) on it. On the other hand, by a remarkable result of Wickstead, there exists a representation approach for every Archimedean vector lattice $E$ in terms of $S(X)$-spaces. In this paper, we show that this representation is order continuous and when $E$ is order complete, it coincides with the known Maeda-Ogasawara representation. Moreover, when $E$ is a Banach lattice, by consideration of the $un$-topology on $E$ and the extended $un$-topology on $S(X)$, we show that this representation is, in fact, a homeomorphism. With the aid of this topological attitude, we establish a representation theorem (in fact a homeomorphism) for the Fremlin projective tensor product between Banach lattices, in terms of $S(X)$-spaces, as well.
Suppose $Σ$ is a topological space and $S(Σ)$ is the vector lattice of all equivalent classes of continuous real-valued functions defined on open dense subsets of $Σ$. In this paper, we establish some lattice and topological aspects of $S(Σ)$. In particular, as an application, we show that the unbounded order convergence and the order convergence are stable under passing to the Fremlin tensor product of two Archimedean vector lattices.
Suppose E and F are locally convex-solid vector lattices. Although we have a suitable vector lattice structure for the tensor product E and F (known as the Fremlin tensor product and denoted by E\otimesF), there is a lack of topological structure on E\otimes F, in general. In this note, we consider a topological attitude on E\otimes F that makes it into a locally convex-solid vector lattice, as well.
Suppose $E$ is a Banach lattice. Recently, there have been some motivating contexts regarding the known Banach-Saks property and the Grothendieck property from an order point of view. In this paper, we establish these results for operators that enjoy different types considered for the Banach-Saks property as well as for different notions related to the Grothendieck property. In particular, beside other results, we characterize order continuity and reflexivity of Banach lattices in terms of the corresponding bounded operators defined on them.
Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent notions for compact operators on $X$. Furthermore, notion of the $AM$-property in $X$ as an extension for the $AM$-spaces in Banach lattices has been considered, recently. In this paper, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices.
In this note, we consider several notions related to the Grothendieck property. Among them, we introduce the notion "unbounded Grothendieck property" in a Banach lattice as an unbounded version of the known Grothedieck property in the Banach space theory. Beside other results, surprisingly, we show that spaces with the unbounded Grothendieck property are exactly the reflexive Banach lattices.
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 orthomomorphisms and the underlying vector lattice to the case of all bounded orthomomorphisms. Furthermore, we investigate topological and ordered structures for these classes of orthomorphisms, as well.
The aim of this note is to consider different notions for the Banach-Saks property in locally solid vector lattices as an extension for the known concepts of the Banach-Saks property in Banach lattices. We investigate relations between them; in particular, we shall characterize spaces in which, these notions agree.
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 ( $uaw$-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded operators between Banach lattices which maps $uaw$-Cauchy sequences to weakly ( $uaw$- or norm) convergent sequences. This allows us to characterize $KB$-spaces and reflexive spaces in terms of these operators, too. Furthermore, we consider the unbounded Banach-Saks property as an unbounded version of the weak Banach-Saks property. There are many considerable relations between spaces possessing the unbounded Banach-Saks property with spaces fulfilled by different types of the known Banach-Saks property. In particular, we characterize order continuous Banach lattices in terms of these relations, as well.
Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce two types of continuous operators between Banach lattices using unbounded absolute weak convergence. We characterize reflexive Banach lattices in terms of these spaces of operators. Furthermore, we investigate whether or not the adjoint of these classes of operators has the corresponding property. In addition, we show that these kinds of operators are norm closed but not order closed. Moreover, we establish how these classes of continuous operators are connected to the well-known classes of $M$-weakly compact operators and $L$-weakly compact operators. Finally, we show that the notions of an $M$-weakly operator and a $uaw$-Dunford-Pettis operator have the same meaning; this extends one of the main results of Erkursun-Ozcan et al. (TJM, 2019).
Suppose $X$ is a locally solid vector lattice. In this paper, we introduce the notion "$AM$-property" in $X$ as an extension for $AM$-spaces in the category of all Banach lattices. With the aid of this concept, we characterize spaces in which bounded sets and order bounded sets agree. This, in turn, characterizes conditions under which each class of bounded operators on $X$ is order bounded and vice versa. Also, we show that under some natural assumptions, different types of bounded order bounded operators on $X$ have the Lebesgue or Levi property if and only if so is $X$.
Motivated by the recent definition of $AM$-property in locally solid vector lattices [O. Zabeti, arXiv: 1912.00141v2 [math.FA]], in this note, we try to investigate those results in the category of all locally solid lattice rings. In fact, we characterize locally solid lattice rings in which order bounded sets and bounded sets agree. Furthermore, with the aid of $AM$-property, we find conditions under that, order bounded group homomorphisms and different types of bounded group homomorphisms coincide. Moreover, we show that each class of bounded order bounded group homomorphisms on a locally solid lattice ring $X$ has the Lebegsue or the Levi property if and only if so is $X$.
Suppose $G$ is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on $G$ which have topological structures. In this paper, our attempt is to assign lattice structures on them. More precisely, we use of a version of the remarkable Riesz-Kantorovich formulae and Fatou property for bounded order bounded homomorphisms to allocate the desired structures. Moreover, we show that unbounded convergence on a locally solid lattice group is topological and we investigate some applications of it. Also, some necessary and sufficient conditions for completeness of different types of bounded group homomorphisms between topological rings have been obtained, as well.
Suppose $X$ is a locally solid lattice ring. It is known that there are three classes of bounded group homomorphisms on $X$ whose topological structures make them again topological rings. In this note, we consider lattice structure on them; more precisely, we show that, under some mild assumptions, they are locally solid lattice rings.
Suppose $(X,Σ,μ)$ is a finite measure space, $E$ is a Banach lattice, and $B(X,E,μ)$ is the space of all Bochner integrable $E$-valued functions. In this note, we show that $B(X,E,μ)$ is a $KB$-space or has the sequential Fatou property if and only if so is $E$. Among this, some results about Bochner integral convergence in $B(X,E,μ)$, using order structure of $E$, have been proved, as well.
In this paper, using the concept of unbounded absolute weak convergence ($uaw$-convergence, for short) in a Banach lattice, we define two classes of continuous operators, named $uaw$-Dunford-Pettis and $uaw$-compact operators. We investigate some properties and relations between them. In particular, we consider some hypotheses on domain or range spaces of operators such that the adjoint or the modulus of a $uaw$-Dunford-Pettis or $uaw$-compact operator inherits a similar property. In addition, we look into some connections between compact operators, weakly compact operators, and Dunford-Pettis ones with $uaw$-versions of these operators. Moreover, we examine some relations between $uaw$-Dunford-Pettis operators, $M$-weakly compact operators, $L$-weakly compact operators, and $o$-weakly compact ones. As a significant outcome, we show that the square of any positive $uaw$-Dunford-Pettis ($M$-weakly compact) operator on an order continuous Banach lattice is compact. Many examples are given to illustrate the essential conditions, as well.