Hereditary Subspaces And Linear Preservers For Minus Partial Order On B(H)

In the last few decades, many researchers have studied the properties of various partial orders on semigroups, such as star partial order, minus partial order and so on. With the extensive and thorough research on matrix and operator algebras, some of these results have been extensively studied on matrix and operator alge-bras. Recently, some preserver problems respect to certain partial order have been considered(cf[1,2]). On the other hand, we know that linear preserver problems are also important. Thus it is interesting to consider those linear maps preserving minus partial order on operator algebras. However, among other things, what are kernels of those maps? It is elementary that those kernels must contain all elements which are dominated by their own elements under this partial order. Hence, motivated by the hereditary subalgebras of C*-algebras on classical partial order on the set of all positive elements(cf[3]), we give the definition of minus partial order-hereditary subspaces on operator algebras and then give a characterization of those subspaces. So we get some results about linear maps preserving minus partial order on B(H). This paper contains three chapters, main content of every chapter as follows:In chapter1, we mainly introduce the background of this paper, some notions, definitions and some results which are always used in this paper.In chapter2, we mainly give the characterizations of minus partial order-hereditary subspaces on operator algebras. In this section, It is proved that a norm closed subspace M. is a minus partial order-hereditary subspace in B(H) if and only if there is a unique pair of projections P and Q in B(H) such that M∩K(H)=PK(H)Q and M—WOT=PB{H)Q, where K{H) is the set of all com-pact operators in B(H) and M—WOT is the weak operator topology closure of M.In chapter3, we mainly discuss about the complexion of linear maps which preserve minus partial order on B(H). By the results obtained in chapter2, it is proved that bounded linear maps preserving minus partial order are injective. Then we obtain a sufficient and necessary condition of elementary operators of length one preserving minus partial order, that is, Let A,B∈B(H) be nonzero operators and △(X)=AXB for a11X∈B(H).Then△is minus partial order-preserving if and Only if both A and B*are bounded below.