Researches On Several New Classes Of Generalized Inverses

The generalized inverse of matrices is a,very important topic in the matrix theory.In 1955,R.Penrose used the four matrices equations to give the concept of the gen-eralized inverse(now we call it Moore-Penrose inverse),and in 1958 M.P.Drazin introduced the definition of the Drazin inverse in rings and semigroups,since then,the generalized inverse of matrices has developed rapidly,and has extensive applications in many diseiplines.These are established mainly for complex matrices,the bounded linear operators on Banach space(or Hilbert space),Banach algebra(or C*-algebra),rings and semigroups.With the development of the theory of generalized inverses,several new classes of generalized inverses were found,such as the Bott-Duffin(e,f)-inverse,core inverse and dual core inverse,and(b,c)-inverse.In this thesis,we mainly adopt some ring theoretic methods to study these new classes of generalized inverses,and obtain some meaningful results.The main contents are arranged as follows:Part 1 mainly studies the Bott-Duffin(e,f)-inverse in rings.First of all,some necessary and sufficient conditions for the existence of the Bott-Duffin(e,f)-inverse of an element in a ring are given by means of invertibility of certain elements.Then,the existence of the Bott-Duffin(e,f)-inverse for a product of three elements is characterized under some prescrilbed conditions,and the relationship between the Bott-Duffin(e,f)-inverse of paq and the Bott-Duffin(e1,f1)-inverse of pa and the Bott-Duffin(e2,f2)-inverse of aq is established.At last,as applications,the existence and the expression of the Bott-Duffin(E,F)-inverse of the 2×2 matrices over R are discussed.Part 2 is concerned with the core inverse and the dual core inverse in*-rings.We study the existence of the core inverse and dual core inverse for a product of three elements under some prescribed conditions.As applications,for the two block matrices(?)and(?)we give the necessary and sufficient conditions for the existences and the expressions of the core inverse and dual core inverse of them when a is core invertible(or d is dual core invertible,resp.)in R,and the case when a E R is invertible.Part 3 mainly investigates the(b,c)-inverse in rings and semigroups.Firstly,the character-ization and representation of the(b,c)-inverse in a*-ring are given,which generalized the related results of image-kernel(p,q)-inverse considered by D.Mosic.Secondly,a new relationship be-tween the(b,c)-inverse and the Bott-Duffin(e,f)-inverse in semigroups is established;that is,if b,c are regular,an element a is(b,c)-invertible if and only if it is Bott-Duffin(bb-,c-c)-invertible,where b-,c-are inner inverses of b and c respectively.Meanwhile,we discuss the existence of the(b,c)-inverse for a product of three elements,and present the relationship be-tween the(b,c)-inverse of paq and the(b',c')-inverse of a.Finally,the existence and expression of the(B,C)-inverse of the lower triangular matrices over rings are given.Especially,we obtain the existence and the expression of the inverse along an element of a lower triangular matrix A over an arbitrary ring;this expression simplifies the result that was investigated by X.Mary and P.Patricio over Dedekind-finite ring.Part 4 considers the reverse order law of(b,c)-inverse in rings and semigroups.First,we present some necessary and sufficient conditions for the existence of the(b,c)-inverse,and give several representations for the(b,c)-inverse related to the group inverse under some pre-scribed conditions.Then,we present equivalent conditions for the reverse order law(a1a2)(b,c)=?2(b,s)?1(t,c)and various mixed-type reverse order laws of the(b,c)-inverse to hold in a semigroup,which generalized the related results studied by H.H.Zhu et al.for the inverse along an element.A more general case of the reverse order law(?1?2)(b3,c3)=?2(b2,c2)?1(b1,c1)is considered too.In Part 5 we introduce new generalized inverses in a ring-one-sided(b,c)-inverse and one-sided annihilator(b,c)-inverse,derived as extensions of(b,c)-inverse by M.P.Drazin.These generalized inverses also generalize one-sided inverse along an element,which was recently intro-duced by H.H.Zhu et al.We investigate the existence,the double commuting and the generalized Cline's formula for these new generalized inverses.