Star Partial Order Hereditary Subspaces And Linear Maps Preserving Star Partial Order On B(H)

Order relation is an important theory of semigroups. For a partial order on B(H), an important property is the heredity of the partial order. In this paper, we firstly define a notion of hereditary subspace with respect to star partial order on B(H), called a hereditary subspace for star partial order. To do this, let "≤" be a partial order on B(H) and M a subspace of B(H), if A∈B(H), B∈M, and A≤B imply A∈M, then we say that M is a hereditary subspace with respect to the partial order "≤". Then, we characterize the characterizations of star partial order-hereditary subspaces in B(H). It is proved that a weak operator topology closed nonzero subspace M in B(H) is hereditary with respect to the star partial order, if and only if there is a unique pair of nonzero projections P and Q in B(H) such that M=PB(H)Q. Besides, in order to study the structures of order algebras, preserver problems are also one of important topics which are concerned by many researchers. Therefore, we lastly discuss the linear maps on K(H) which preserve the star partial order. Let H be a separable infinite-dimensional complex Hilbert space and let K(H) be the set of all compact operators on H. It is proved that if φ is a linear bijection map on K(H) preserving the star partial order, then there exist a nonzero α∈C, two unitary or anti-unitary operators U and V on H such that φ(X)=αUXV for all X∈K(H) or φ(X)=αU X*V for all X∈K(H).