The Research On The Core Inverse And Dual Core Inverse

Generalized inverses can be classified into classical generalized inverses and new types of generalized inverses.The classical generalized inverses includes Moore-Penrose inverse and Drazin inverse(it is called group inverse when Drazin index is 1),these two kinds of generalized inverses play important roles in many fields,for examples,differential equations numerical analysis,optimization,electrical network analysis,Markov chains and metrology Some new generalized inverses have emerged in recent years,such as core inverse and dual core inverse,which are introduced by Baksalary and Trenkler in 2010.In 2014,Rakic et al.generalized the core inverse of a complex matrix to the case of an element in a ring with involution,and they used five equations to characterize the core inverse.In 2017,Xu et al.proved that five equations for the core inverse are equivalent to three equations.Up to now there is not so much research on the core inverse and dual core inverse.This paper mainly studies characterizations and expressions of the core inverse and dual corer inverse in rings semigroups and categories,and then we apply these results to some special matricesAs two classical generalized inverses,Moore-Penrose inverse and group inverse play important roles in the theory of generalized inverses,many scholars devoted themselves to studying their existences and expressions.Bhaskara Rao characterized the existence of group inverse by idempotents,Xu et al.characterized the existence of Moore-Penrose inverse by projections The second section of the Chapter 2 generalized these results to the core inverse and dual core inverse,and we characterize the existence of the core inverse and dual core inverse by projections.We prove that for any positive integer n?1,a is core invertible if and only if there exists a unique projection(or a Hermitian element)p such that pa=0 and an+p is invertible and we give the expression for the core inverse of a.As we all know,we can characterize the Moore-Penrose inverse by the intersection of a principal left ideal and a principal right ideal,Zhu et al.transformed this characterization to one principal ideal,that is to say,a is Moore-Penrose invertible if and only if ????*?R if and only if a E R??*?.The first section of the Chapter 2 prove that for any positive integer k?2,? is core invertible if and only if ??R(?*)k? ? R?k.Compared with the above conclusion of Zhu et al.,we show that a is both core invertible and dual core invertible if and only if ? ? ?(?*)k?R?R?(?*)k?.Note that a regular element is also Moore-Penrose invertible in C*-algebra,but it is not valid in a ring.So it is interesting to investigate existences of generalized inverses for regular elements in rings.Chen et al.discussed the core invertibility of a regular element in a ring,and they gave characterizations for a regular element which was both core invertible and dual core invertible.The third section of the Chapter 2 characterize the core inverse and dual core inverse of a regular element by its inner inverse and invertible elements,and show some new characterizations which are different from results of Chen et al..The generalized inverse of morphism is one of the important contents in the generalized inverse theory.For examples,Robinson and Puystjens investigated the Moore-Penrose invertibility and group invertibility of a morphism with factorization or kernel and cokernel.Huylebrouk and Puystjens,You and Chen considered the Moore-Penrose inverse and group inverse of a sum of morphism in categories.And they applied conclusions to the ring case and showed relations between the Moore-Penrose invertibility and group invertibility of ?+j and a,where a is a element in ring R and j is a Jacobson radical of R.Chapter 3 obtain corresponding results for the core inverse,we establish the relations between the core invertibility of ?+j and a.And we prove that ?+j is core invertible if and only if(1-??(?))j(1+?(?)j)-1(1-?(?)?)=0,the expression for the core inverse of a+j is also gave.In addition,we discuss the core and dual core invertibility of a morphism with factorization or kernel and cokernel,and we give characterizations and expressions for the core and dual core inverse.The calculation of generalized inverse of some special matrices(such as companion matrix,Hankel matrix,etc.)plays an important role in matrix theory.Hartwig and Shoaf studied the Drazin inverse of bidiagonal matrices,triangular Toeplitz matrices and triangular Toeplitz matrices,they investigated the relations between the group invertibility of the companion matrix L of the polynomial p(?)and the coefficient of p(?).Motivated by them,We discuss the relations between the core invertibility of the companion matrix L and the coefficient of p(?)in Chapter 4.In addition,we investigate the core inverse and expressions of Hankel matrix,Toeplit matrix and Bezout matrix by means of matrix decomposition.The existence of the generalized inverse of the difference and product of two projection operators(or idempotent operators)on a ring or Hilbert space has attracted the attention of many scholars.Li,Deng and Wei investigated existences and expressions of the Moore-Penrose inverse of the difference and product of two projections in C*-algebra and bounded linear operator of Hilbert space.Zhang,Chen and Zhu et al.studied characterizations and expressions of the Moore-Penrose inverse(Drazin inverse)of the difference and product of two projections in rings.The last chapter consider characterizations and expressions of the core inverse of p-q and 1-qp,pq+qp,repectively,where p and q are two projections.In addition Guo,Castro-Gonzalez and Hartwig studied the forward order law for {1}-inverse and Moore-Penrose inverse for complex matrices.Compared with their results,we investigate equivalent conditions for the forward order laws and hybrid forward order laws to hold for the core inverse.