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A short proof on lifting of projection properties in Riesz spaces

Marek Wójtowicz (1999)

Commentationes Mathematicae Universitatis Carolinae

Let L be an Archimedean Riesz space with a weak order unit u . A sufficient condition under which Dedekind [ σ -]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 ( X ) -spaces. Similar results are obtained for the Riesz spaces B n ( T ) , n = 1 , 2 , , of all functions of the n th Baire class on a metric space T .

AM-Compactness of some classes of operators

Belmesnaoui Aqzzouz, Jawad H'michane (2012)

Commentationes Mathematicae Universitatis Carolinae

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

Daugavet centers and direct sums of Banach spaces

Tetiana Bosenko (2010)

Open Mathematics

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...

Equations containing locally Henstock-Kurzweil integrable functions

Seppo Heikkilä, Guoju Ye (2012)

Applications of Mathematics

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.

Generalized M-norms on ordered normed spaces

I. Tzschichholtz, M. R. Weber (2005)

Banach Center Publications

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 -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...

Monotone extenders for bounded c-valued functions

Kaori Yamazaki (2010)

Studia Mathematica

Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, C ( 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 ( A , c ) 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...

M-weak and L-weak compactness of b-weakly compact operators

J. H'Michane, A. El Kaddouri, K. Bouras, M. Moussa (2013)

Commentationes Mathematicae Universitatis Carolinae

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).

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 class of positive almost weak Dunford-Pettis operators

Abderrahman Retbi (2015)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we introduce and study the class of almost weak Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP property. Next, we characterize pairs of Banach lattices for which each positive almost weak Dunford-Pettis operator is almost Dunford-Pettis.

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