| The Drazin inverses of matrices and operators are very important for the study of matrix theory and operator theory. Since Drazin introduced the concept of Drazin inverse, Drazin inverses have played essential roles in theoretical research and have been indispensable tools in many application fields.Let R be an associative ring. For a∈R, if there exists x∈R such that ax=xa, xax=x, a-a2x is nilpotent, then x is said to be the Drazin inverse of a, written x=ad. If nilpotency is replaced by quasi-nilpotency, then x is said to be the generalized Drazin inverse of a.The theory, applications and computational methods of Drazin inverses have been developed rapidly during the last few decades. The Drazin inverse is a subject with great theoretical interests and applications in several areas, including Markov chains, differential equations, statistics, numerical analysis, population models, cryptography and control theory. Interesting topics of Drazin inverse include the explicit representations for the Drazin inverses of block matrices, the generalized Drazin inverses of operator matrices, the generalized Drazin inverses of sums and the weight Drazin inverses of block matrices, and computation and perturbation bounded of Drazin inverses.In this dissertation, we focus on the explicit representations for the Drazin inverses of block matrices. the generalized Drazin inverses of operator matrices and the generalized Drazin inverses of sums.In Chapter 1, we give a comprehensive survey on the representations for the Drazin inverses of block matrices, operator matrices and sums.In Chapter 2, we present the explicit representations for the Drazin inverses of block matrices. Let M=(?) be a 2×2 block complex matrix, where A∈Cm×m, D∈Cn×n. It is of interest to derive explicit expressions for the Drazin inverse of the matrix M in terms of the blocks. This problem is very important in many research fields but is not yet solved completely. Many authors have considered this open problem under extra conditions. We give the explicit representation for the Drazin inverses of M under the condition that BDiC=0, i=0,1,…,n-1. As an application, we give the explicit representation for the Drazin inverses of the sum of two matrices P, Q under the condition that PQiP=0,i=1,…,n. Based on these results, the explicit representation for the Drazin inverses of M is presented under the condition that ABC=0, BDiC=0, i= 0,1,…, n. Our results generalize many results in literature. 0, i=0,1,…,n-1, then ind(A)≤ind(M)≤ind(A)+2ind(D). ind(A)=r, ind(D)=s and k≥max{ind(M),ind(D)}. If BDiC=0, i=0,1,…,n-1, then where Theorem 2.3.1 Let P and Q∈Cn×n, ind(P)=r, ind(Q)=s and k=r+2s. If PQiP=0, i=1, 2,…,n, then where then where X1, Y1 and Z1 are correspondingly X, Y and Z in Theorem 2.2.2 with A, B, C, D replaced by A2 + BC, AB + BD; CA + DC, CB + D2, respectively; U1, V1, W1 are correspondingly U, V, W in Theorem 2.3.1 with P, Q replaced by CB, D2, respectively; In Chapter 3, we study the following problem for matrices of bounded linear operators on Banach spaces:For a 2×2 operator matrix M=(?), where A∈B(X), D∈B(Y), B∈B(Y,X) and C∈B(X,Y), when has M the generalized Drazin inverse and in this case how to represent the generalized Drazin inverse of M in terms of the blocks? a solution of this problem is useful for countable Markov chains, abstract Cauchy problem, infinite dimensional linear singular differential equations, iteration procedure.First, we give the explicit representation for the generalized Drazin inverses of M under the condition that BDd=0, BDiC=0, i=0,1,2,…. Then the representation for the generalized Drazin inverses of M is obtained under the condition that ABC= 0, BDd=0 (or DdC=0), BDiC=0, i=1,2,…. Finally, we consider several special cases of our main results, which generalized recent results.Theorem 3.2.1 Let A and D be generalized Drazin invertible. If BDd=0, BDiC=0, i=0,1,2, then M be generalized Drazin invertible and whereTheorem 3.3.1 Let A and D be generalized Drazin invertible. If BDd=0, DπCA=0, DπCB=0, then M be generalized Drazin invertible and whereThe last chapter is devoted to representations of the generalized Drazin inverses for the sum of two elements in Banach algebras and the sum of two bounded linear operators on Banach spaces.Theorem 4.1.1 Let a, b∈A be generalized Drazin invertible. If aπb=b, abπ=a, bπbaaπ=0, then a+b be generalized Drazin invertible, andTheorem 4.2.1 Let P, Q∈B(X) be generalized Drazin invertible. If QQπP=0 and PQd=0, then P+Q be generalized Drazin invertible, andTheorem 4.2.2 Let P, Q∈B(X) be generalized Drazin invertible. If PQP=0 and Q2P=0, then P+Q be generalized Drazin invertible, and... |
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