The Order And Topology Properties Of Disjointness Preserving Operators

Disjointness preserving operators(D.P.O)are important special operators on Riesz space.The thesis discusses the problems about the range of D.P.O and the order and topology properties of D.P.O.In the first part,it's devoted to the characterization of range space of ordered D.P.O and the images description.Firstly,it's proven that the equivalence condition,which it's range space is Riesz subspace,is that T E is contained in |T|E or |T|E is contained in T E.Then based on other people's researches, it's right that when |T| is interval preserving operator,the image of principal ideal effected by D.P.O T is principal ideal.And when D.P.O T is surjective and E has principal projection property,the image of principal band effected by T is principal band.In the second part,the order properties of D.P.O are discussed.Firstly,I summarize the conditions that the sum of two D.P.O also is D.P.O,and then give another description of one of them,which is that when the infimum of |T| and |S| isn't zero,the sum of D.P.O T and D.P.O S is D.P.O if and only if the sum of T₂ and S₂ is D.P.O,where T₂ and S₂ are the components of T and S on the disjoint complement of the infimum of T and S.Secondly,based on the factorization property of regular operators space,It's proven that n-disjoint operator is the sum of n disjointness preserving operators which are mutually disjoint.Finally,I give an conclusion that the elements of the band generated by finite D.P.O are all D.P.O if the sums of arbitrary two of finite D.P.O are D.P.O.In the last part,on the base of my supervisor's research of the closeness of principal bands in lattices of operators,some conditions can be reduced. Supposed E and F be Banach lattice with F Dedekind completed,If the operator T from E to F is D.P.O,the principal band generated by T in the regular operators spaces is closed in the sense of operator norm.And the set of all D.P.O also is closed.