Research On The Topic Of Core Partial Order And Dual Core Partial

Let H be an infinite-dimensional complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. In this paper, we give various char-acterizations of core partial order and dual core partial order, and the relationship among the core partial order, the dual core partial order and the other partial order. Then, use the method of operator block matrices, we present the infimum A?(?) B of A and B with respect to given orders. Furthermore, definitions and properties of the core, the dual core, the left core and the right core partial orders are reseaxsched in ring. The main results are as follows:1.Let A,B in B~1(H),A?(?) B if and only if there exist project operator P, idempotent operator Q in B(H), such that A=PB=BQ=QA.2.Let A-B in B~1(H), if AB=BA, then A B if and only if B-A? B3.Let A, B in B~1(H), A?B(?) B if and only if A?- B, A*B is self-adjoint, BA2=ABA.4.Let A, B?B~1(H),V(A, B)=: {M(?)QH:M(?)QR(A)?R(B), PMAPmA = APMA, PMBPMB= BPMB,PMA=PMB}.the infimum A?(?)B of and B is exist, and A?(?)B = PM0A=PM0B, M0 is maximal element in P(A, B).5.Let R be a ring with the unit 1, (?)a,b?R(?), The following axe equivalent:(1)a?(?)b;(2)a=aa+b, a=ba(?)a;(3)There exist self-adjoint idempotent p?R, idempotent p?R such that oa=op?ao=go?oa=oq, pa=pb, aq=bq;(4)There exist self-adjoint idempotent p?R, idempotent p?R such that a, b have following form:...