Additive Maps Preserving Operator Range Or Kernel Inclusion

The preserving problem is one of important research fields of operator algebra theory. Range and kernel are also the basic concepts of linear operator. Since this paper from the two basic concepts of linear operator to analysis and research the operator algebra on two important preserving problem:the characterization of additive mappings preserving range or kernel inclusion, the characterization of additive maps preserving ascent of operators less than or equal.Firstly, Let B(X) be the Banach algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We define a map φ:B(X)→B(X) preserving range or kernel inclusion in both direction, and define a map φ:B(X)→B(X) preserving ascent of operators less than or equal, we obtain φ satisfied those conditions is a bijection.Secondly, we prove successively the map φ preserving range or kernel inclusion in both direction also preserving surjectivity or injectivity of operators, the set of operators of rank one in both directions, and φ(Ⅰ) is invertible, we obtain that the structure characterization of the maps φ,φ(T)=UTV, VT∈B(X).Finally, we study the characterization, of additive map preserving ascent of operators less than or equal by the definition and properties of ascent.