B. Aqzzouz

We establish a sufficient condition under which the product of an order bounded almost Dunford-Pettis operator and an order weakly compact operator is Dunford-Pettis. And we derive some consequences.

We generalize a result on the duality property for order weakly compact operators and use it to establish some characterizations of positive operators.

We give some sufficient conditions under which the linear span of positive compact (resp. Dunford-Pettis, weakly compact, AM-compact) operators cannot be a vector lattice without being a sublattice of the order complete vector lattice of all regular operators. Also, some interesting consequences are obtained.

We define the $\varepsilon_{\infty }$-product of a Banach space $G$\ by a quotient bornological space $E\mid F$ that we denote by $G\varepsilon _{\infty }(E\mid F)$, and we prove that $G$ is an $% \mathcal{L}_{\infty }$-space if and only if the quotient bornological spaces $G\varepsilon _{\infty }(E\mid F)$ and $% (G\varepsilon E)\mid (G\varepsilon F)$ are isomorphic. Also, we show that the functor $\mathbf{.\varepsilon }_{\infty }\mathbf{.}:\mathbf{Ban\times qBan\longrightarrow qBan}$ is left exact. Finally, we define the $\varepsilon _{\infty }$-product of a b-space by a quotient bornological space and we prove that if $G$ is an $% \varepsilon $b-space\ and $E\mid F$ is a quotient bornological space, then $(G\varepsilon E)\mid (G\varepsilon F)$ is isomorphic to $G\varepsilon _{\infty }(E\mid F)$.

We define a space of holomorphic functions $O_{1}(U,E/F)$, where $U$ is an open pseudo-convex subset of $\Bbb{C}^{n}$, $E$ is a b-space and $F$ is a bornologically closed subspace of $E$, and we prove that the b-spaces $O_{1}(U,E/F)$ and $O(U,E)/O(U,F)$ are isomorphic.