We study the relationship between weak* Dunford–Pettis and weakly (resp. M-weakly, order weakly, almost M-weakly, and almost L-weakly) operators on Banach lattices. The following is one of the major results dealing with this matter: If E and F are Banach lattices such that F is Dedekind $\sigma $-complete, then each positive weak* Dunford–Pettis operator $T:E\rightarrow F$ is weakly compact if and only if one of the following assertions is valid: (a) the norms of $E^{\prime }$ and F are order continuous; (b) E is reflexive; and (c) F is reflexive.

First we give a counterexample showing that recent results on separate order continuity of Arens extensions of multilinear operators cannot be improved to get separate order continuity on the product of the whole of the biduals. Then we establish conditions on the operators and/or on the underlying Riesz spaces/Banach lattices so that the extensions are order continuous on the product of the whole biduals. We also prove that all Arens extensions of any regular multilinear operator are order continuous in at least one variable and we study when Arens extensions of regular homogeneous polynomials on a Banach lattice $E$ are order continuous on $E^{**}$.

We show that for the case of uniformly convex Banach spaces, the conditions of Brøndsted’s fixed point theorem can be relaxed.

We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $\text{Ces}_{\infty }$ and its sequence counterpart $\text{ces}_{\infty }$ are isomorphic. This is rather surprising since $\text{Ces}_{\infty }$ (like Talagrand’s example) has no natural lattice predual. We prove that $\text{ces}_{\infty }$ is not isomorphic to $\ell _{\infty }$ nor is $\text{Ces}_{\infty }$ isomorphic to the Tandori space $\widetilde{L_{1}}$ with the norm $\Vert f\Vert _{\widetilde{L_{1}}}=\Vert \widetilde{f}\Vert _{L_{1}}$, where $\widetilde{f}(t):=\text{ess}\,\sup _{s\geqslant t}|f(s)|$. Our investigation also involves an examination of the Schur and Dunford–Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro–Marcinkiewicz and Cesàro–Lorentz spaces have the latter property.

We prove that Krivine’s Function Calculus is compatible with integration. Let $(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ be a finite measure space, $X$ a Banach lattice, $\mathbf{x}\in X^{n}$, and $f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ a function such that $f(\cdot ,\unicode[STIX]{x1D714})$ is continuous and positively homogeneous for every $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$, and $f(\mathbf{s},\cdot )$ is integrable for every $\mathbf{s}\in \mathbb{R}^{n}$. Put $F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$ and define $F(\mathbf{x})$ and $f(\mathbf{x},\unicode[STIX]{x1D714})$ via Krivine’s Function Calculus. We prove that under certain natural assumptions $F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$, where the right hand side is a Bochner integral.

We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an “almost disjointness preserving” operator is well approximable by a disjointness preserving one. However, this approximation is not always possible, as our examples show.

In this short note, we correct and reformulate Theorem 3.1 in the paper published in Proceedings of the Edinburgh Mathematical Society58(3) (2015), 617–629.

There are two main aims of the paper. The first is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second is to extend the criterion for the precompactness of sets in the Lebesgue spaces Lp (ℝn ), 1 ⩽ p < ∞, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces , into Lorentz-type spaces.

In this paper we use the norm of bounded variation to study multilinear operators and polynomials on Banach lattices. As a result, we obtain when all continuous multilinear operators and polynomials on Banach lattices are regular. We also provide new abstract M- and abstract L-spaces of multilinear operators and polynomials and generalize all the results by Grecu and Ryan, from Banach lattices with an unconditional basis to all Banach lattices.

In this paper we give conditions under which a positive order continuous operator T defined on a Banach function space can be extended, preserving the order continuity in a certain optimal way. The optimal domain for T turns out to be a space of weakly integrable functions with respect to a vector measure (defined on a δ-ring) canonically associated to T. A similar result is obtained when T is σ-order continuous and we want to preserve the σ-order continuity. We apply these results to kernel operators.

Following ideas used by Drewnowski and Wilansky we prove that if $I$ is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and $I$ does not contain any order copy of ${{\mathbb{R}}^{\mathbb{N}}}$ then there exists a closed, separable, discrete Riesz subspace $G$ such that the topology induced on $G$ is Lebesgue, $I\,\bigcap \,G\,=\,\left\{ 0 \right\}$, and $I\,+\,G$ is not closed.

It is proved that if a positive operator $S\,:\,E\,\to \,F$ is $\text{AM}$-compact whenever its adjoint ${{S}^{'}}:{{F}^{'}}\to {{E}^{'}}$ is $\text{AM}$-compact, then either the norm of $\text{F}$ is order continuous or $E\prime $ is discrete.

We study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.

Let $A={{({{a}_{j,k}})}_{j,k\ge 1}}$ be a non-negative matrix. In this paper, we characterize those $A$ for which ${{\left\| A \right\|}_{E,F}}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: ${{\ell }_{p}}$, $d(w,p)$, and ${{\ell }_{p}}(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

We prove the converse of a theorem of Zaanen about the duality problem of positive $\text{AM}$-compact operators.

We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

The main result in this paper is a representation theorem for homogeneous orthogonally additive polynomials on Banach lattices. The representation theorem is used to study the linear span of the set of zeros of homogeneous real-valued orthogonally additive polynomials. It is shown that in certain lattices every element can be represented as the sum of two or three zeros or, at least, can be approximated by such sums. It is also indicated how these results can be used to study weak topologies induced by orthogonally additive polynomials on Banach lattices.

Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón–Lozanovskiĭ construction. Factorization theorems for operators in spaces more general than the Lebesgue ${{L}^{p}}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt ${{A}_{p}}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from ${{L}^{p}}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón–Lozanovskiĭ construction are involved in the proofs.

New criteria and Banach spaces are presented (for example, $GL$-spaces and Banach spaces with property $(\alpha)$) that ensure that the Boolean algebra generated by a pair of bounded, commuting Boolean algebras of projections is itself bounded. The notion of $R$-boundedness plays a fundamental role. It is shown that the strong operator closure of any $R$-bounded Boolean algebra of projections is necessarily Bade complete. Also, for a Dedekind $\sigma <formula form="inline" disc="math" id="frm006"><formtex notation="AMSTeX">$-complete Banach lattice $E$, the Boolean algebra consisting of all band projections in $E$ is $R$-bounded if and only if $E$ has finite cotype. In this situation, every bounded Boolean algebra of projections in $E$ is $R$-bounded and has a Bade complete strong closure.

We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that l1, c0 and l∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class ∧. We also present examples of weighted Orlicz spaces with the Schur property which are not L1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.