In this paper, we give some necessary and sufficient conditions such that each positive operator between two Banach lattices is weak almost Dunford-Pettis, and we derive some interesting results about the weak Dunford-Pettis property in Banach lattices.

The present paper is devoted to some applications of the notion of L-Dunford-Pettis sets to several classes of operators on Banach lattices. More precisely, we establish some characterizations of weak Dunford-Pettis, Dunford-Pettis completely continuous, and weak almost Dunford-Pettis operators. Next, we study the relationships between L-Dunford-Pettis, and Dunford-Pettis (relatively compact) sets in topological dual Banach spaces.

Let $L$ be an Archimedean Riesz space with a weak order unit $u$. A sufficient condition under which Dedekind [$\sigma$-]completeness of the principal ideal $A_u$ can be lifted to $L$ is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of $C\left(X\right)$-spaces. Similar results are obtained for the Riesz spaces $B_n\left(T\right)$, $n=1,2,\cdots$, of all functions of the $n$th Baire class on a metric space $T$.

We introduce new concept of almost demi Dunford–Pettis operators. Let $E$ be a Banach lattice. An operator $T$ from $E$ into $E$ is said to be almost demi Dunford–Pettis if, for every sequence $\left\{x_n\right\}$ in $E_{+}$ such that $x_n\to 0$ in $\sigma (E,E^{\text{'}})$ and $\parallel x_n-T{x}_n\parallel \to 0$ as $n\to \infty$, we have $\parallel x_n\parallel \to 0$ as $n\to \infty$. In addition, we study some properties of this class of operators and its relationships with others known operators.

We characterize Banach lattices on which each regular order weakly compact (resp. b-weakly compact, almost Dunford-Pettis, Dunford-Pettis) operator is AM-compact.

A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X 1⊕F X 2 of two Banach spaces X 1 and X 2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X 1 and X 2 there exists a Daugavet center acting from X 1⊕F X 2, and the class of those F such that for some pair of spaces X 1 and X 2 there is a Daugavet center...

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.

In ordered Banach algebras, we introduce eventually and asymptotically positive elements. We give conditions for the following spectral properties: the spectral radius belongs to the spectrum (Perron--Frobenius property); the spectral radius is the only element in the peripheral spectrum; there are positive (approximate) eigenvectors for the spectral radius. Recently such types of results have been shown for operators on Banach lattices. Our results can be viewed as a complement, since our structural...

L-norms and M-norms on Banach lattices, unit-norms and base norms on ordered vector spaces are well known. In this paper m- and $m_{\le }$-norms are introduced on ordered normed spaces. They generalize the notions of the M-norm and the order-unit norm, possess also some interesting properties and can be characterized by means of the constants of reproducibility of cones. In particular, the dual norm of an ordered Banach space with a closed cone turns out to be additive on the dual cone if and only if the...

Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, $C_{\infty }(A,c)$ the set of all bounded continuous functions f: A → c, and $C_A(X,c)$ the set of all functions f: X → c which are continuous at each point of A ⊂ X. We show that a Tikhonov subspace A of a topological space X is strong Choquet in X if there exists a monotone extender $u:C_{\infty }(A,c)\to C_A(X,c)$. This shows that the monotone extension property for bounded c-valued functions can fail in GO-spaces, which provides a negative answer to a question...

The paper deals with the properties of a monotone operator defined on a subset of an ordered Banach space. The structure of the set of fixed points between the minimal and maximal ones is described.

We characterize Banach lattices under which each b-weakly compact (resp. b-AM-compact, strong type (B)) operator is L-weakly compact (resp. M-weakly compact).