A note on weak almost limited operators

Authors : Nabil MACHRAFİ, Kamal EL FAHRİ, Mohammed MOUSSA, Birol ALTIN

Pages : 759-770

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Publication Date : 2019-06-15

Article Type : Research Paper

Abstract :Let us recall that an operator $T:E\rightarrow F,$ between two Banach lattices, is said to be weak* Dunford-Pettis insert ignore into journalissuearticles values(resp. weak almost limited); if $f_{n}\leftinsert ignore into journalissuearticles values( Tx_{n}\right); \rightarrow 0$ whenever $insert ignore into journalissuearticles values(x_{n});$ converges weakly to $0$ in $E$ and $insert ignore into journalissuearticles values(f_{n});$ converges weak* to $0$ in $F^{\prime }$ insert ignore into journalissuearticles values(resp. $f_{n}\leftinsert ignore into journalissuearticles values( Tx_{n}\right); \rightarrow 0$ for all weakly null sequences $\leftinsert ignore into journalissuearticles values( x_{n}\right); \subset E$ and all weak* null sequences $\leftinsert ignore into journalissuearticles values(f_{n}\right); \subset F^{\prime }$ with pairwise disjoint terms);. In this note, we state some sufficient conditions for an operator $R:G\rightarrow E$insert ignore into journalissuearticles values(resp. $S:F\rightarrow G$);, between Banach lattices, under which the product $TR$ insert ignore into journalissuearticles values(resp. $ST$); is weak* Dunford-Pettis whenever $T:E\rightarrow F$ is an order bounded weak almost limited operator. As a consequence, we establish the coincidence of the above two classes of operators on order bounded operators, under a suitable lattice operations` sequential continuity of the spaces insert ignore into journalissuearticles values(resp. their duals); between which the operators are defined. We also look at the order structure of the vector space of weak almost limited operators between Banach lattices.
Keywords : weak almost limited operator, weak Dunford Pettis operator, weak Dunford Pettis property, Banach lattice