Research On The Bounds And The Order Topology For The Diamond Partial Order In B(H)

Let H be a complex Hilbert space with dim H≥ 2 and B(H)the algebra of all bounded linear operators on H.Let ≤◇ be the diamond order on B(H),that is,for A,B∈B(H),we say that A ≤◇ B if (?) and AA*A=AB*A.In this paper,we mainly consider the minimal upper bounds and maximal lower bounds for the subsets in B(H)with respect to the diamond partial order.At the same time,we discuss the order topology under this partial order.The main results are as follows.In the first part,we focus on the relevant properties for the operators in B(H)satisfying the diamond partial order.Based on the special properties of some oper-ators under such partial order,we describe the forms of the minimal upper bounds for a bounded set in B(H).Meanwhile,we discuss the convergence of the increasing operator nets and give the sufficient and necessary conditions to have the supremum for the nets.Besides,for the maximal lower bounds and the decreasing nets,we do some similar work.In the second part,we discuss the relevant properties of the order topology under the diamond partial order.Furthermore,we compare such topology with other topologies in B(H),especially the order topology under the*-partial order,which greatly enriches the topology properties in B(H).