We study the class of Dunford-Pettis sets in Banach lattices. In particular, we establish some sufficient conditions for which a Dunford-Pettis set is relatively weakly compact (resp. relatively compact).

We investigate the sufficient condition under which each positive b-weakly compact operator is Dunford-Pettis. We also investigate the necessary condition on which each positive b-weakly compact operator is Dunford-Pettis. Necessary condition on which each positive b-weakly compact operator is weakly compact is also considered. We give the operator that is semi-compact, but it is not bweakly. W...

3. N. Dunford and B. J. Pettis, Linear operators on summable functions, Trans. Amer. Math. Soc. vol. 47 (1940) pp. 323-392. 4. O. Nikodym, Sur une generalisation des integrates de M. Radon, Fund. Math, vol. 15 (1930) pp. 137-179. 5. R. S. Phillips, Integration in convex linear topological space, Trans. Amer. Math. Soc. vol. 48 (1940) pp. 516-540. 6. C. E. Rickart, Integration in a convex linear...

In the standard theory of delay equations, fundamental solution does not ‘live’ in state space. To eliminate this age-old anomaly, we enlarge As a consequence, lose strong continuity operators and this, turn, has as consequence that Riemann integral no longer suffices for giving meaning to variation-of-constants formula. compensate, develop Stieltjes-Pettis setting norming dual pair spaces. Par...

Let X(i), i t [0; 1] be a collection of identically distributed and pairwise uncorrelated random variables with common finite mean # and variance a 2. This paper shows the law of large numbers, i.e. the fact that ~ X ( i ) d i = #. It does so by interpreting the integral as a Pettis-integral. Studying Riemann sums, the paper first provides a simple proof involving no more than the calculation o...

Let A be the disk algebra, Ω be a compact Hausdorff space and μ be a Borel measure on Ω. It is shown that the dual of C(Ω, A) has the Dunford-Pettis property. This proved in particular that the spaces L(μ,L/H 0 ) and C(Ω, A) have the Dunford-Pettis property.

A new characterization of the Dunford-Pettis property in terms of the extensions of multilinear operators to the biduals is obtained. For the first time, multilinear characterizations of the reciprocal Dunford-Pettis property and Pe lczyński’s property (V) are also found. Polynomial and holomorphic versions of these properties are given as well.

We introduce a weakened version of the Dunford-Pettis property, and give examples of Banach spaces with this property. In particular, we show that every closed subspace of Schreier’s space S enjoys it. As an application, we characterize the weak polynomial convergence of sequences, show that every closed subspace of S has the polynomial Dunford-Pettis property of Biström et al. and give other p...