### ${\mathcal{L}}_{\pi}$-Spaces and cone summing operators

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Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $${H}_{\varphi}\left(\mu \right){\tilde{\otimes}}_{l}Y$$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $$\left({H}_{\varphi}\left(\mu \right){\tilde{\otimes}}_{l}Y\right)*$$ and $${H}_{\varphi}\left(\mu \right)*{\tilde{\otimes}}_{l}Y*$$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým...

We give a characterization of conditional expectation operators through a disjointness type property similar to band-preserving operators. We say that the operator T:X→ X on a Banach lattice X is semi-band-preserving if and only if for all f, g ∈ X, f ⊥ Tg implies that Tf ⊥ Tg. We prove that when X is a purely atomic Banach lattice, then an operator T on X is a weighted conditional expectation operator if and only if T is semi-band-preserving.

For Banach lattices X with strictly or uniformly monotone lattice norm dual, properties (o)-smoothness and (o)-uniform smoothness are introduced. Lindenstrauss type duality formulas are proved and duality theorems are derived. It is observed that (o)-uniformly smooth Banach lattices X are order dense in X**. An application to an optimization theorem is given.

We complete a result of Hernandez on the complex interpolation for families of 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.

We study an order boundedness property in Riesz spaces and investigate Riesz spaces and Banach lattices enjoying this property.

For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator ${T}_{s}$, 0 < s < 2, to be the linear extension of the map $\left({h}_{I}\right)/{\left(\right|I|}^{1/s})\mapsto \left({h}_{\tau \left(I\right)}\right)\left(\right|\tau \left(I\right){|}^{1/s})$, where ${h}_{I}$ denotes the ${L}^{\infty}$-normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that ${T}_{s\u2080}$ is bounded on ${H}^{s\u2080}$, then for all 0 < s < 2 the operator ${T}_{s}$ is bounded on ${H}^{s}$.

The Coifman-Fefferman inequality implies quite easily that a Calderón-Zygmund operator T acts boundedly in a Banach lattice X on ℝⁿ if the Hardy-Littlewood maximal operator M is bounded in both X and X'. We establish a converse result under the assumption that X has the Fatou property and X is p-convex and q-concave with some 1 < p, q < ∞: if a linear operator T is bounded in X and T is nondegenerate in a certain sense (for example, if T is a Riesz transform) then M is bounded in both X and...