The Research On The Drazin Inverse Of Elements In Rings And The Generalized Drazin Inverse Of Elements In Banach Algebras

Drazin inverse is a crucial branch of generalized inverse,which has a wide range of applications in many fields,such as numerical analysis,singular value decomposition,surveying,cryptography and Markov chains.Since it was introduced,a host of scholars have conducted research on the Drazin inverse in Banach algebra,the Drazin inverse for bounded linear operators on Hilbert spaces,and the Drazin inverse in rings and semigroups.Although fruitful results have been achieved,there are still a great number of issues which need to be further explored.This article mainly focuses on the explicit representations for the Drazin inverse of elements over a ring,the generalized Drazin inverse of sums,the generalized Drazin inverse of block matrices and operator matrices,and it consists of four parts:(1)We give a comprehensive research on the representations for the Drazin inverse of elements over a ring,the generalized Drazin inverse of sums,block matrices and operator matrices.(2)We mainly examine the explicit representations for the Drazin inverse of the sum,the difference and the product of elements over a ring.To begin with,the sufficient and necessary conditions for the existence and the formulae of the Drazin inverse of a+b which are obtained,where a and b both are Drazin invertible elements satisfying aba=a2b and bab=b2a.Then,the formula for the Drazin inverse of ab is considered under the conditions aba=λa2b=λ’ba2 and bab=μb2a=μ’ab2(λ,λ’,μ,μ’ are nonzero complex numbers);meanwhile the equivalent conditions for the existence and the representations of the Drazin inverse of a-b are given.(3)We will mainly extend the relevant conclusions of(2)from rings to Banach algebras.Firstly,we consider the existence criteria of the generalized Drazin inverse of the sum of two generalized Drazin invertible elements under the conditions ab2=bab,ba2=aba,and the explicit representations of a+b are given.Furthermore,we consider conditions on a and b with multiplicative perturbations so that the sum will have generalized Drazin inverse.Secondly,the equivalent conditions for the existence and the formulas of the generalized Drazin inverse of a+b are obtained,where a and b are generalized Drazin invertible elements satisfying aba=a2b and bab=b2a.Finally,under the conditions with parameters,the existence and expressions of the generalized Drazin inverse of linear combination of a,b are obtained.(4)As an application of additive results in(3),we first obtain the expressions for the generalized Drazin inverse of a partitioned matrix(?)with the generalized Schur complement s=d-cadb nonsingular and the blocks satisfying sca=0,aπb=0;meanwhile,we give the equivalent conditions for the existence of the group inverse of(?).Secondly,the representations of the generalized Drazin inverse of(?)is obtained,when the generalized Schur complement s=d-cadb=0 as well as the blocks satisfying aπbc=bcaπ,aaπbc=0.In the end,we obtain the formulae for the generalized Drazin inverse of the operator matrix(?)with the blocks fulfilling certain circumstances.Some conclusions in this current article not only generalize but also unify many previous results.