Unbounded continuous operators and unbounded Banach–Saks property in Banach lattices

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms weakly null nets, we introduce unbounded operators replacing weak convergence with absolutely (uaw-convergence) lattices. We characterize order lattices and reflexive these operators. Moreover, motivated characterizing lattice Cauchy sequences, consider pre-unbounded which maps uaw-Cauchy sequences to (uaw- or norm) convergent sequences. This allows us KB-spaces operators, too. Furthermore, Banach–Saks property as an version property. There are many relations possessing fulfilled different types known In particular, relations, well.