Research On Projections, Generalized Inverses And Effect Algebras In B(H)

Projections, generalized inverses and quantum effects are heated topics in operator theory and also have important value in both theory and application. The research of these subjects has related to pure and applied mathematics such as algebra, geometry, perturbation theory, matrix analysis, approximation theory, multivariate linear modelling, Banach-algebra, numerical analysis, optimality principle and quantum physics etc. Through research, the internal relations and constructions among operators can be found and a substantial basis can be provided for the study of the theory of operators.The research of this thesis focuses on common complements of two subspaces, the Moore-Penrose inverse of projections, Drazin inverse of operators, factorizations and infimum of quantum effects. The research on projections and generalized inverses brings out the following results: the geometric representation of projections on Hilbert space, the representation of Moore-Penrose inverse of sum and product of two projections, and that of Dazin inverse and the optimal approximation of the reduced minimum modulus of Drazin inverse. The research on quantum effects on Hilbert space contains factorization problem and the infimum problem of Hilbert space effects. This article is divided into five chapters.In chapter 1, by using space decomposition, block operator matrices in different decomposition of Hilbert space, the common completion problem of two closed sub-spaces are treated completely. The operator matrix representations of orthogonal projections onto two closed subspaces are given. Using these representations, we have given a construction proof on common complement problem and give a complete answer to Groβ's question. These block operators matrix methods not only make proof clearer, but also give us more information about geometrical structures between two subspaces.In chapter 2, the idempotent operators on infinite dimensional Hilbert spaces are discussed. By using the operator matrix representations of orthogonal projections which had been given in the first chapter, the new characterization of gap and conorm between two subspaces of a Hilbert space are established. Secondly, we give the expressions of the minimum gap and the angle between two closed subspaces by the properties of the projections on these subspaces. At last, by using block operator matrix representation of idempotent operators, we prove that the invertibility of linear combination of two idempotents is independent of the choice of coefficients if the sum of these two coefficients is not zero.In chapter 3, Moore-Penrose inverse on Hilbert spaces are treated completely. By using block operator matrices, the polar decomposition and space decomposition, we have established the Moore-Penrose inverse of products and differences of...