Some New Generalized Inverses And Partial Order Of Bounded Operators

The research of generalized inverses theory mainly involves complex matrices,bounded linear operators on Banach(Hilbert)spaces,matrices over rings and morphisms in a category.Classical generalized inverses such as Moore-Penrose inverses and Drazin inverses(group inverses)play an important role in many fields such as differential equation,numerical analysis,cybernetics and so on.In recent years,with the development of generalized inverse theory,core inverses,pseudo core inverses,generalized(pseudo)Drazin inverses and*-DMP elements have been introduced.Based on the bounded linear operators on complex Hilbert space and complex Banach algebra,several new generalized inverses and partial orders on complex Banach algebras mentioned above are deliberated.The main conclusions are as follows:In the first part,we study the core inverses,the pseudo core inverses and the*-DMP elements of bounded linear operators on complex Hilbert spaces.(1)If A,E?B(H),A has some generalized inverses A?,and I+A?E is invertible,the necessary and sufficient conditions under which(A+E)? exists and(A+E)?=(I+A?E)-1 A? are studied,where ? could be(1,3),(1,3,6),(1,3,7),(?).(2)If Tk=PAQ is generalized factorization with A is ?1,4}-invertible,the necessary and sufficient conditions under which T is pseudo core invertible with ind(T)?k are achieved;if A is ?1}-invertible(or Moore-Penrose invertible),the necessary and sufficient conditions under which T is*-DMP element are obtained.The results of Gao et al.on GDH-factorization are extended to generalized factorization,and the condition that A is projection is weakened to that A is ?1?-invertible or A is Moore-Penrose invertible.In the second part,generalized Drazin inverses and pseudo Drazin inverses of elements on the complex Banach algebra A are studied.It is shown that:(1)a? A is generalized Drazin invertible if and only if a=a+J? A/J is generalized Drazin invertible;a?A is pseudo Drazin invertible if and only if a=a+J?A/J is Drazin invertible.Thus,a new idea to study pseudo Drazin invertibility of elements on Banach algebras by using Drazin inverse is presented.As an application,it is proved that if P,Q?B(X)are idempotent operators,?,?? C\{0} and ? ? C,then aP+?Q-?PQ is pseudo Drazin invertible if and only if P-Q is pseudo Drazin invertible if and only if P+Q is pseudo Drazin invertible.Benabdi and Barraa's latest results on Drazin inverses are generalized to pseudo Drazin inverses.(2)The pseudo Drazin order is in(?)troduced in A,and it is proved that the pseudo Drazin order is preorder.At the same time,the necessary and sufficient conditions under which elements in Banach Algebras have pseudo Drazin order are presented.