Moore-penrose Inverse. Hilbert Space Densely Defined Closed Operator Perturbation Theorem

The theory of generalized inverses has wide applications, and has become an important branch in modern mathematics. Generalized inverse theory has many subobjects, such as generalized inverses of matrix, the generalized inverses of linear transformations in linear space, the linear generalized inverses of linear operator in Hilbert space, orthogonal generalized inverse, the linear generalized inverses of linear operator in Banach space. The theory of generalized inverses has its genetic roots essentially in the context of the so-called"ill-posed"linear problems. These include problems in which one either specifies too much information, or too little. These problems cannot be solved in the sense of a solution of a nonsingular problem. However, there is a sense in which there is still a solution (and in fact a unique"solution") if one adopts, for example, the notion of"least-squares solution"or"minimal norm solution"and so on. As a result, those who meet with"ill-posed"problem of linear subjects, giving rise to a generalized inverse. Combining the generalized inverse with non-linear analysis tools, we can also solve a large number of non-linear"ill-posed"problems. Therefore, the theory of generalized inverse is of great theoretical significance and has applications in practice.In 1920, E. H. Moore introduced the concept of generalized inverse matrix. R. Penrose showed in 1955 that there exists a unique matrix B satisfying the following four relations: ABA = A, BAB = B, ( AB )*= AB and ( BA) *= BA. These conditions are exactly equivalent to Moore's. The unique matrix B satisfying these relations is now known as the Moore-Penrose inverse, and is denoted by . Since the Moore-Penrose generalized inverse possesses the least-squares property, its continuity has been studied extensively. In recent years, the Moore-Penrose generalized inverse of bounded linear operators in Hilbert space has been investigated by J. Ma, W. Cao, G. Song, G. Chen, Y. Xue, M. Wei, Y. Wei, Q. Huang and others. And many continuity characterizations of the Moore-Penrose generalized inverse have been obtained. It should be noted that these characterizations are studied in the case of bounded linear operator. How to discuss the continuity of the Moore-Penrose generalized inverse of unbounded linear operator? Since the domain of unbounded linear operator is not the whole space, the techniques used in the case of bounded linear operator can not be directly applied to the case of unbounded linear operator. New techniques and methods must be introduced. This paper deals with the perturbed problems of Moore-Penrose generalized inverse of closed linear operator in Hilbert space:We know that the densely defined and closed linear operator is an important class of unbounded linear operator. Let T be a closed linear operator from X into Y such that its domain is dense in X and T has the bounded Moore-Penrose generalized inverse . It is nature to investigate the following perturbed problem: What condition on the"small"perturbationδT can guarantee that the Moore-Penrose generalized inverse of the perturbed operator exists? If it exists, can we give an explicit expression of As far as we know, the condition of the norm of smaller than 1 is always assumed in appeared papers. If the norm of smaller than 1 is assumed, then the invertibility of the operator and the boundedness of the inverse operator can be obtained directly from the well-known Banach Lemma. Then how to discuss the corresponding perturbation problems without assuming the norm of smaller than 1? Therefore, the key to consider such problem is how to prove that the operator is invertible and its inverse operator is bounded.In this paper, using a new method, we prove the invertibility of the operator . Then based on this result, we obtain the sufficient conditions for the Moore-Penrose generalized inverse of operator T being stable, that is:Let X and Y be two Hilbert spaces. Let T∈C ( X , Y) with and with the bounded Moore-Penrose generalized inverse . LetδT∈L ( X , Y) be T - bounded with the T - bound b smaller than 1, i.e. If satisfies Then R (T ) is closed and T = T +δT has bounded Moore-Penrose generalized inverse T ?, and Where PN ( T) is the unique norm-preserving extension of on X .