A. Elbour

The aim of this paper is to investigate the relationship between limited operators and weakly compact (resp. compact) operators. Mainly, it is proved that if every limited operator $T:E\rightarrow X$ from a Banach lattice $E$ into Banach space $X$ is weakly compact (resp. compact) then the norm of $ E^{\prime }$ is order continuous or $X$ has the (BD) property (resp. GP property). Also, it is proved that if every weakly compact operator $ T:E\rightarrow X$ is limited then the norm of $E^{\prime }$ is order continuous or $X$ has the DP$^{\ast }$ property.

Метою цієї роботи є дослідження зв'язку між обмежувальними операторами та слабо компактними (відповідно компактними) операторами. Доведено, що якщо кожен обмежувальний оператор $T : E\rightarrow X$ з банахової ґратки $E$ в банаховий простір $X$ є слабо компактним (відповідно компактним), то норма в $E^{\prime }$ є порядково неперервною або $X$ має (BD)-властивість (відповідно GP-властивість). Також доведено, що якщо кожний слабо компактний оператор $T : E\rightarrow X$ обмежений, то норма в $E^{\prime }$ є порядково неперервною або $X$ має DP$^{\ast }$-властивість.

This paper is devoted to investigation of conditions on a pair of Banach lattices $E; F$ under which every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited. Mainly, it is proved that if every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, then the norm on $E'$ is order continuous or $F$ is finite dimensional. Also, it is proved that every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, if one of the following statements is valid:
1) The norm on $E^{\prime }$ is order continuous, and $F^{\prime }$ has weak$^{\ast }$ sequentially continuous lattice operations.
2) The topological dual $E^{\prime }$is discrete and its norm is order continuous.
3) The norm of $E^{\prime }$ is order continuous and the lattice operations in $E^{^{\prime }}$ are weak$^{\ast }$ sequentially continuous.
4) The norms of $E$ and of $E^{\prime }$ are order continuous.

We give some new characterizations of almost weak Dunford-Pettis operators and we investigate their relationship with weak Dunford-Pettis operators.

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.