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On invariant elements for positive operators.

R. Zaharopol (1997)

Revista Matemática de la Universidad Complutense de Madrid

In the paper we study the existence of nonzero positive invariant elements for positive operators in Riesz spaces. The class of Riesz spaces for which the results are valid is large enough to contain all the Banach lattices with order continuous norms. All the results obtained in earlier works deal with positive operators in KB-spaces and in many of them the approach is based upon the use of Banach limits. The methods created for KB-spaces cannot be extended to our more general setting; that is...

On order structure and operators in L ∞(μ)

Irina Krasikova, Miguel Martín, Javier Merí, Vladimir Mykhaylyuk, Mikhail Popov (2009)

Open Mathematics

It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.

On set-valued cone absolutely summing maps

Coenraad Labuschagne, Valeria Marraffa (2010)

Open Mathematics

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space ℒ 1 ∑ , c b f ( X ) of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of ℒ 1 ∑ , c b f ( X ) , and to derive necessary...

On stability and robust stability of positive linear Volterra equations in Banach lattices

Satoru Murakami, Pham Ngoc (2010)

Open Mathematics

We study positive linear Volterra integro-differential equations in Banach lattices. A characterization of positive equations is given. Furthermore, an explicit spectral criterion for uniformly asymptotic stability of positive equations is presented. Finally, we deal with problems of robust stability of positive systems under structured perturbations. Some explicit stability bounds with respect to these perturbations are given.

On the class of order almost L-weakly compact operators

Kamal El Fahri, Hassan Khabaoui, Jawad H'michane (2022)

Commentationes Mathematicae Universitatis Carolinae

We introduce a new class of operators that generalizes L-weakly compact operators, which we call order almost L-weakly compact. We give some characterizations of this class and we show that this class of operators satisfies the domination problem.

On the class of order Dunford-Pettis operators

Khalid Bouras, Abdelmonaim El Kaddouri, Jawad H'michane, Mohammed Moussa (2013)

Mathematica Bohemica

We characterize Banach lattices E and F on which the adjoint of each operator from E into F which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if E and F are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator T from E into F has an adjoint Dunford-Pettis operator T ' from F ' into E ' if, and only if, the norm of E ' is order continuous or F ' has the Schur property. As a consequence we show that, if E and F are two Banach...

On the equality between some classes of operators on Banach lattices

Belmesnaoui Aqzzouz, Aziz Elbour, Mohammed Moussa (2012)

Mathematica Bohemica

We establish some sufficient conditions under which the subspaces of Dunford-Pettis operators, of M-weakly compact operators, of L-weakly compact operators, of weakly compact operators, of semi-compact operators and of compact operators coincide and we give some consequences.

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