Idempotent Operators Are Indeed Bound In Decreasing Order

Let P and Q be idempotents on a Hilbert space H.The minus partial order P<Q is equivalent to PQ=QP=P.In the set of idempotent operators.If the supremum and infimum of set of operator P and Q with respect to the minus partial order exist,we use P(?)Q and P(?)Q respectively to denote the minimum upper bound and a maximum lower bound of P and Q with respect to the order?.We use Qo?and Qor respectively to denote the maximum value of the orthogonal projection smaller than Q and the minimum value of the orthogonal projection larger than Q with respect to the order ?.he related properties of P(?)Q,P(?)Q,Qo? and Qo? are mainly studied in this paper.The main contents are as follows:In Chapter one,we introduce some basic concepts of symmetric operator,Jprojection,Krein space,operator order,star order,minus order and some lemmas to be used in this paper.In Chapter two,we first study the effect of minus order P?Q on operator order P<Q and the equivalent condition of P?Q and P(?)Q.Secondly,the structural characteristics corresponding to the satisfaction when the value of P(?)Q?P(?)Q is a specific operator are studied.Finally,we explore the properties of P(?)Q when P and Q are interchangeable with the symmetric operator J.In Chapter three,we first study the structural characteristics of Qor and Qor,and generalizes the similar results to J-projection.If Q is a J-projection,then Qor and Qor are also J-projection.And if P E P(H)is a J-projection,the equivalent condition of finding J-projection Q ? B(H)Id/{P(H)} making Qor=P or Qor=P is given.Secondly,the relationship between P(?)Q and P0r(?)Qor and the necessary and sufficient conditions of Por? Qor are studied.Finally,we explored the weak operator topology limit of sequence ?Qnor?n?N.