Advances In Sum Of Projections

This paper is a review of the articles, after reading the related documents carefully, I made a simple report concerning some results about the sum of projections from the later period of last century to the beginning of this century and developed a discussion about the problem of characterizing Hilbert space operators which are of a sum of projections.The first section of this paper gives some simple definitions and notations.Definition 1.1. An operator T is positive (resp.strictly positive),denoted by T≥θ(resp.T＞θ)if(Tx,x)≥0(resp.(Tx,x)＞0)Definition 1.2. An orthogonal projection is an operator P with P2= P= P*.Definition 1.3. For Hermitian operators A and B, then A≥B(resp. A＞B) if A-B≥8(resp. A-B＞θ).For an operator T on H and 1≤n≤∞, T(n) denotes the operator The trace of T,when defined,is denoted by tr T, and the range and the rank of T are ran T and rank T.The first result of this section is how to characterize sum of projections in terms of a certain operator matrix representation.Proposition 1.1. If T is a strictly positive operator. Then T is a sum of projections iff T(?)θis unitarily equivalent to an operator matrix of the form whereθand I1,…,In denote the zero and identity operators on spaces.Note that in the preceding proposition the sufficient part is satisfied even assuming only the positivity of T.In the following, it introduces an important theorem.Theorem 1.1. A complex square matrix A is a sum of projections iff A≥θ, tr A is an integer, and tr A≥rank A.For the remaining part of section 1, we just take operators on infinite-dimensional sep-arable spaces into consideration, and gives the definition of essential norm.Definition 1.4. let||T||e denote the essential norm of T:||T||e= inf{||T+K||:K compact}.After full preparation it gives the last theorem in this section.Theorem 1.2. Any positive operator with essential norm strictly greater than one is the sum of projections.The first important theorem in section 2 is concerning a necessary condition for such operators to be sums of projections.Theorem 1.3. Let T= I+K,where K is an infinite-rank compact operator, and it is also a sum of projections, then both K+and K-have infinite rank.By the way, it gives the definition of the positive and negative parts of a Hermitian operator A.Definition 1.5. respectively, whereIt then gives a wonderful result about Hermitian operators.Theorem 1.4. A Hermitian operators A can be written as A=aP+bQ with projections P,Q in generic position and nonzero real a,b iff for some real c the following conditions are satisfied:(a)A-cI and-(A-cI) are unitarily equivalent,(b)either 0＜|A-cI|＜|c|I or|c|I＜|A-cI| is valid. The final result in this section is about the trace about sums of projections of the form identity+compact.Theorem 1.5. IfT= I+K, where K is infinite-rank compact. Suppose T is injective and a sum of projections, then either tr K+= tr K_=∞or K is of trace class with tr K a negative integer.Section 3 is about some results of the convex combination of projections, it divides into 2 parts to introduce, part 1 of which contains two main results.Proposition 1.2. Let T be an n×n matrix satisfying O≤T≤I.(1)Then T admits an expression as a convex combination of'n+1' commuting projec-tions.(2)Ifn＞k are positive integers and tr T= k, then T admits an expressions as a convex combination of'n'commuting rank-k projections.The'quoted'number of commuting projections in each expression is sharp in the sense that it is the smallest integer for the statement to be valid.Theorem 1.6. A positive contraction A with finite spectrum o-(A) is a convex combination of projections.Moeroverσ(A) consists of rational numbers if and only if A is an average of commuting projections.The main theorem in part 2 is.Theorem 1.7. For the smallest integer m such that every n×n matrix T satisfying O≤T≤I, let p(n) be the smallest m. Then p(n)=[log2n]+2.The last section gives a general summarization about the first three sections, and clas-sifies the sum-of-projections problem simply by some variations. It begins with the case of sums of commuting projections.Proposition 1.3. T is the sum of infinite-rank projections if and only if T itself has infinite rank and is the sum of projections.The last theorem of this section is also the last one of this paper, which is very interest-ing.Theorem 1.8. Rank is weakly lower semicontinuous. In fact, the stronger result is also true.corollary 1.1. Rank is strongly lower semicontinuous.