The Inverse Of Disjointness Preserving Operators On Banach Lattice

The bijective disjointness preserving operators are the central object of this work . Let T: X → Y be a bijective disjointness preserving operator from a vector lattice X to a vector lattice Y .Is T-1 also a disjointness preserving operator ? For arbitrary Riesz space, the problem is not affirmative. Now , this is an open problem.To a large extent , the article shows some additional conditions on operator that would guarantee the affirmative solutions to this problem and some counterexamples in their absence.The main results are followed:1) Characterization Of the inverse of disjointness preserving operators.One hand , we discuss the characterization of the inverse of disjointness preserving operators on the classical Banach lattices; the other hand, by use of condition (a), we also discuss the characterization of the inverse of disjointness preserving operators on the Riesz spaces .2) The article is devoted to condition (a), a principal technical tool in this work. The condition (a) is a rather natural generalization of condition (β) .But they are independent.3)The article introdunces two types of new operators which are discussedabout the inverse of disjointness preserving operators.4) Disdcuss an important subclass of disjointness preserving operators: lattice homomorphism operators. The article mainly talks about the characterization of lattice homomorphism operators on the classical Banach lattices and these...