Solution Of Generalized Inverse And Operator Equations

Operator equation is one of the key contents of the operator theory. Due to the extensive application in cybernetics, dynamic program, statistics etc and so on,the study of operator equation make a great progress in recent years.As an important branch of modern mathematics, the theory of generalized inverse plays an important role in many fields such as differential equation, numerical algebra, linear statistical inference, optimization and Markov chain.In this paper, we use the generalized inverse to study some issues of operator equation, for the existence conditions of the reduced solution and Douglas solution of operator equation AX=B, and also we give the expression of the generalized inverse for general solution in Banach spaces and Hilbert spaces. Based on this, we use the perturbation theory of generalized inverse to investigate the continuity of Douglas solution.This paper contains three chapters. The first chapter is devoted to the introduction and preliminary, In chapter 2, we utilize perturbation theory of generalized inverse to study the existence of the reduced solution and Douglas solution of operator equation. Using the generalized inverse theory of densely defined closed operator, we investigate the continuity of solution of operator equation in chapter 3. The main results in this paper are as follows:Theorem 1 Let H, K and G be Banach spaces, A∈C(H,K), B∈B(G,K) and A+∈C(K,H) be the generalized inverse of A, then there is a solution to operator equation AX= B if and only if R(B) (?) R(A). In this case, A+B is a solution to AX=B, and the set of all the solutions to equation AX=Bis {A+B+(I-A+A)V, V ∈B(D(A), K)}.Theorem 2 Let H, K and G be Banach spaces, A e C(H,K), B ∈ B(G,K), and suppose that N(A) has the topological complement M in H. H= N(A)(?)M, then the derived solution of equation AX = B satisfying R(X)(?)M must be A+B, where A+ is a generalized inverse of A with respect To H=N (A) (?)M.Theorem 3 Let H, K and G be Hilbert spaces, A∈C(H,K) and B∈B(G,K) . If R(B) (?) R(A), A(?)∈C(K, H) is the Moore-Penrose generalized inverse of A, then A(?)B is the unique solution to operator equation AX = B satisfying R(X)(?)R(A*), which is called the Douglas solution of AX=B.Theorem 4 Let A∈C(H,K), B∈B(G,K), and ΔAn∈B(H,K).ΔAnâ†'0, if An=A + ΔAn satisfying R(B)(?)R(An), R(B)(?)R(A), R(A) and R(An) is closed. Moreover then the Douglas solution A\B of perturbed equation AnX = B converges to the Douglas solution A(?)B of AX = B.Theorem 5 Let A∈C(H,K), B∈B(G,K), Bn∈B(G,K).And Bnâ†'B,ΔAnâ†'0,if An=A + ΔAn Satisfying R(B)(?)R(A), R(Bn) (?)R(An),R(A) and R(An) is closed and sup ||An||<∞, then the Douglas solution An(?)Bn of perturbed equation AnX = Bn converges to the Douglas solution A(?)B of AX=B.