The Research On (b,c)-inverse And Related Generalized Inverses

Moore-Penrose inverse and Drazin inverse are two important generalized inverses. There are some important applications in various fields. Many scholars obtained a large amount of results in complex matrices, Banach algebra and the bounded linear operators on Hilbert spaces. In 2012, M.P. Drazin introduced the (b, c)-inverse in rings and semigroups, unified the Moore-Penrose inverse, Drazin inverse and other classical generalized inverses. It provided a new public platform for the research of generalized inverses. Due to this, the research of the (b, c)-inverse becomes more difficult. Recently, the related study on (b, c)-inverse is not rich, there are still many problems waiting for further investigations. In this thesis, we mainly focus on the Moore-Penrose inverse, Drazin inverse and (b, c)-inverse to further the study of the invertibility of linear combinations, the existence of Moore-Penrose inverse of block matrices, reverse order laws of generalized inverses, the existence of the (b, c)-inverse and the (b,c) spectrum idempotents. The thesis consists of four parts.The first part, first of all, the necessary and sufficient conditions for b is Moore-Penrose invertible are given whenever a*≤b and a is Moore-Penrose invertible, for two given elements a and b in a ring. This result generalized the relevant conclusion of C.Y. Deng on the bounded linear operators. Secondly, under the condition of partial orderings, the Moore-Penrose invert-ibility of linear combinations of two Moore-Penrose invertible elements are discussed, these results generalized the relevant conclusions of M. Tosic on the invertibility of the linear combi-nations of EP elements and generalized projectors. Finally, a new criterion for the existence of Moore-Pemose inverse of product matrices is obtained. As applications, the necessary and suf-ficient conditions for the existence of Moore-Penrose inverse of block matrices M=(ab cd) where a is invertible and M=(ab od) are given, which extend the work of R.E. Hartwig and P. Patricio.The second part, by using the extremal ranks of generalized Schur complements, we discuss the mixed-type reverse order laws as the form (AB)I = BI(AIABBI)IAI, where A, B are two complex matrices,I={1,3},{1,2,3},{1,3,4}. Using the rank of matrices, the sufficient and necessary conditions are established for these mixed-type reverse-order laws of {1,3}-、{1,2,3}- and {1,3,4}-inverse. The results enrich the research of mixed-type reverse order laws.The third part, using ring-theoretical methods and skills, we discuss the Drazin invertibil-ity of the sums and products of two Drazin invertible elements. Firstly, the Drazin invertibility of a+b is given, for two elements a and b satisfied ab= λba in algebras over an arbitrary field. Meanwhile, using the Drazin invertibility of the elements in corner rings, some proofs on the Drazin invertibility of the sums and products of idernpotents obtained by P. Patricio and J.L. Chen are simplified.The fourth part, we mainly discuss the characterization of the existence of (b,c)-inverse and the (b, c) spectrum idempotent. First of all, we start with a special (b, c)-inverse (Bott-Duffin (e,f)-inverse). Using the invertible elements, the existence and representations of Bott-Duffin (e,f)-inverse are given whenever e and f are two projectors. Secondly, using the annihi-lator, direct decomposition and the invertible elements, a new characterization of the existence of (b, c)-inverse is obtained. Meanwhile, we find that if a is (b, c)-invertible, then both b and c must be regular. Moreover, it is proved that when b and c are regular, (b, c)-inverse, hybird (b, c)-inverse and annihilator (b, c)-inverse are coincident. Finally, elements with equal (b, c) spectrum idempotents related to their (b, c)-inverses are characterized, and the reverse order laws for (b, c)-inverse are considered. Some results on image-kernal inverse given by M. Dijana are generalized.