Perturbation Theory Of Some Generalized Inverse And Its Representation

In this thesis, we mainly discuss the following parts:1. Stable perturbation of bounded linear operators on Hilbert spaces.We mainly investigate the equivalent conditions of stable perturbations for bounded linear operators on Hilbert spaces and characterize invertibility of I+T(?)δT by the range and null spaces of T and T. As applications, we get the representations of the Moore-Penrose inverse of certain2×2operator matrices under some conditions.2. Perturbation analysis for AT,S(2) inverse.By virtue of the gap between subspaces, we investigate the perturbations for AT,S(2) inverse on Hilbert spaces and Banach spaces, respectively. We give the explicit formulas for the pertur-bation of AT,S(2) when T, S and A have perturbations. At the same time, the new results on the upper bounds of‖AT’,S’(2)‖and‖AT’,S’(2)—AT,S(2)‖are obtained.3. Perturbation analysis for group inverse in unital ring.We mainly investigate the perturbation of the group inverse a#on unital (?). Under the stable perturbation, we obtain the explicit expressions of a#. As an application, we give the representation of the group inverse of the matrix on the ring (?) for certain d, b, c∈(?).4. Perturbation analysis for generalized inverse of closed operators.Let T be a closed operators, ST be a linear operators. We mainly investigate the invertiblity of I+δTT+and characterize the stable perturbation for generalized inverse of closed operators on Banach spaces. On Hilbert spaces, we give the expression for the Moore-Penrose inverse of T when T(?) is a bounded linear operators. At the same time, the upper bounds of‖T(?)‖and‖T(?)-T(?)‖are obtained.5. The reverse order law for the Moore-Penrose inverse of closed operators.The reverse order law for the Moore-Penrose inverse of closed linear operators with closed range is investigated by virtue of the Norm-Preserving extension of the bounded linear operators.6. The expression for the Moore-Penrose of A-EF*.Let K, H be Hilbert spaces. Let A be a bounded linear operator on H and E, F be bounded linear operators from K to H with R(A) closed and R(E)(?) R(A), R(F)(?) R(A*). Suppose that EF*A(?)EF*=EF*. Then the Moore-Penrose inverse (A-EF*)(?) of A-EF*is expressed as...