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A description of Banach space-valued Orlicz hearts

Coenraad Labuschagne, Theresa Offwood (2010)

Open Mathematics

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 ϕ μ ˜ 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 H ϕ μ ˜ l Y * and H ϕ μ * ˜ 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...

A disjointness type property of conditional expectation operators

Beata Randrianantoanina (2005)

Colloquium Mathematicae

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.

A dual property to uniform monotonicity in Banach lattices.

W. Kurc (1993)

Collectanea Mathematica

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.

A note on complex interpolation of Banach lattices

Anita Tabacco Vignati, Marco Vignati (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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

A note on L-Dunford-Pettis sets in a topological dual Banach space

Abderrahman Retbi (2020)

Czechoslovak Mathematical Journal

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.

A note on Riesz spaces with property- b

Ş. Alpay, B. Altin, C. Tonyali (2006)

Czechoslovak Mathematical Journal

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

A remark on extrapolation of rearrangement operators on dyadic H s , 0 < s ≤ 1

Stefan Geiss, Paul F. X. Müller, Veronika Pillwein (2005)

Studia Mathematica

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 ( h I ) / ( | I | 1 / s ) ( h τ ( I ) ) ( | τ ( I ) | 1 / s ) , where h I denotes the L -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 is bounded on H s , then for all 0 < s < 2 the operator T s is bounded on H s .

A₁-regularity and boundedness of Calderón-Zygmund operators

Dmitry V. Rutsky (2014)

Studia Mathematica

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...

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