In the present article, we expose various properties of unbounded absolutely weak Dunford-Pettis and unbounded absolutely weak compact operators on a Banach lattice E . In addition to their topological and lattice properties, we investigate relationships between M -weakly compact operators, L -weakly compact operators, and order weakly compact operators with unbounded absolutely weak Dunford-Pettis operators. We show that the square of any positive uaw -Dunford-Pettis ( M -weakly compact) operator on an order continuous Banach lattice is compact. Many examples are given to illustrate the essential conditions.

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Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences c and d on a vector lattice, we study d-asymptotic properties of operator nets formed by c-continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on d-martingale and d-Lotz-Räbiger nets.) is said to be a convergence vector lattice. Let J be an order dense ideal, see [1,2] for more on this notion, in the vector lattice (X, c). Following [5], we write x λwe write uc instead of u J c. Prime examples of c are order convergence and norm convergence which are respectively denoted by o and B Niyazi Anıl Gezer

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