Left Star Order Of Idempotent Operators

The partial order is an important research field in functional analysis.In recent years,the study of the left-star order has attraced the attention of many domestic and foreign scholars.Let A and B be an idempotent operator on a Hilbert space H,the left-star order A*?B is equivalent to A*A=A*B and A=BA.In the set of idempotent operators,if the supremum and infimum of set of operator A and B with respect to the left-star order exist,we use As?B?A*?B to denote the minimun*? upper bound and maximum*?lower bound of A and B respectively.It is well known that(Q(H),*?)is a partially ordered set.In this paper,we mainly study some related properties of idempotents with respect to the left-star order and the relationship between them and the star order;Secondly,on the basis of reference[19,21],some conclusions about J-projection are obtained.The main contents are as follows:In Chapter 1,we mainly introduce the symbols,concepts and theorems commonly to be used in this paper.In Chapter 2,we first give the equivalent representation of operator matrix form of A?B,A*?B,A?*B and A