Research On Some Invariance Properties Of Operator Product

Let H, K, L and M be Hilbert spaces, B(H), B(K, H) denote theset of all bounded linear operators on H and from H into K, respectively. Forgiven A∈B(H, K), B∈B(H, L), C∈B(M, L), and let the range R(B) of B beclosed. Then B has the Moore-Penrose inverse, that is, there is a unique operatorX∈B(L, H) satisfied the following equations. (3)Let B{i,j,…, k} the set of all operators X∈B(L, H) satisfied the (i), (j),…, (k)equationss of the above four equations and written by B^{(i,j,…,k)}. When the set{i, j,…, k} contains the number 1, then B^{(i,j,…,k)} is called a (i, j,…, k) generalizedinverse of B. Generally, the generalized inverse of an operator are not unique.In the recent years, the problem of invariance properties of a triple matrixproduct involving generalized inverses has been observed by many authors, such asJ. K. Baksalary, Jürgen Grob, Yongge Tian, R. Kala, T. Pukkila and so on (see [1-9]). In this article, we mainly study the problems of invariance properties of a tripleoperator product involving generalized inverses, which generalized some results ofJürgen Groβand Yongge Tian (2006) obtained in [12] and J.K. Baksalary and A.Markiewicz (1996) obtained in [10].There are three chapters in this article, and the main content of each chapteras follows:In the first chaper, an alternative proof of the equivalence of Drazin invertibilityof operators AB and BA is given by the Riesz functional calculus. As an application,we will prove thatσ_D(AB)=σ_D(BA) andσ_D(A)=σ_D(?), whereσ_D(M) and (?)denote the Drazin spectrum and the Aluthge transform of an operator M∈B(H),respectively.In the second chapter, we mainly study given three operators A∈B(H, K),B∈B(H, L), C∈B(M, L), and the range R(B) of B is closed, certain invariance properties of the triple operator product AXC with respect to the choice of Xare investigated by the operaior matrix technique, where X is a different type ofgeneralized inverses of B. This generalizes the results obtained by Jürgen GroβandYongge Tian in [12] to the infinite dimenional Hilbert spaces. It is worthily to pointout that our using methods are different from that by Jürgen Groβand YonggeTian.And in the chapter three, we explore for three operators A∈B(H, K), B∈B(H, L), C∈B(K, L), if the range R(B) of B is closed, then the necessary andsufficient conditions such that (?)σ(AB⁽¹⁾C)=C (4)has been obtained by using block-operator matrix technique, whereσ(D) is thespectrum of an operator D∈B(H) and B^{1} is the {1}-inverse of B. It is worthilyto point out that not only we got the necessary and sufficient conditions such that(4) holds, but also we point out the sufficient conditions obtained in the TheoremB-M of [10] are just the necessary and sufficient conditions. Here, our using methodsare different from that used in [10].