The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevc generalized the core inverse of a complex matrix to the case of an element in a ri...The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a,b ∈ R. Then a ∈ R with a= b if and only if (ab)^* = ab, ba^2 = a, and ab^2 = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.展开更多
In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-...In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.展开更多
In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related t...In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to AT,S^(2).展开更多
We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations.This relation becomes a pre-order.Then,one-sided preorders associ...We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations.This relation becomes a pre-order.Then,one-sided preorders associated to the weighted core-EP inverse are given from two perspectives.Finally,we make a comparison for these two sets of one-sided weighted pre-orders.展开更多
We first prove that if a is both left(6,c)-invertible and left(c,b)-invertible,then a is both(6,c)-invertible and(c,6)-invertible in a*-monoid,which generalizes the recent result about the inverse along an element by ...We first prove that if a is both left(6,c)-invertible and left(c,b)-invertible,then a is both(6,c)-invertible and(c,6)-invertible in a*-monoid,which generalizes the recent result about the inverse along an element by L.Wang and D.Mosic[Linear Multilinear Algebra,Doi.org/10.1080/03081087.2019.1679073],under the conditions(a6)*=ab and(ac)*=ac.In addition,we consider that ba is(c,fe)-invertible,and at the same time ca is(6,c)-invertible under the same conditions,which extend the related results about Moore-Penrose inverses studied by J.Chen,H.Zou,H.Zhu,and P.Patricio[Mediterr J.Math.,2017,14:208]to(6,c)-inverses.As applications,we obtain that under condition(a2)^(*)=a^(2),a is an EP element if and only if a is one-sided core invertible,if and only if a is group invertible.展开更多
In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of 1−pq,1−pqp,p−pq and p−q,where p,q are projections in different settings,such as∗-rings,∗-reducing rings and C∗-algebras.More...In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of 1−pq,1−pqp,p−pq and p−q,where p,q are projections in different settings,such as∗-rings,∗-reducing rings and C∗-algebras.Moreover,several representations for the core inverses of product,difference and sum of two generalized projections are derived.In particular,a number of examples are given to illustrate our results.展开更多
In this paper,we mainly give characterizations of EP elements in terms of equations.In addition,a related notion named a central EP element is defined and investigated.Finally,we focus on characterizations of a genera...In this paper,we mainly give characterizations of EP elements in terms of equations.In addition,a related notion named a central EP element is defined and investigated.Finally,we focus on characterizations of a generalized EP element,i.e.,+-DMP element.展开更多
基金Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11201063, 11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020), the Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1501048B), and the Natural Science Foundation of Jiangsu Province (No. BK20141327).
文摘The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a,b ∈ R. Then a ∈ R with a= b if and only if (ab)^* = ab, ba^2 = a, and ab^2 = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.
基金supported by the National Natural Science Foundation of China(Nos.11971294,11801124)China Postdoctoral Science Foundation(No.2020M671068)the Natural Science Foundation of Anhui Province(No.1808085QA16)。
文摘In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.
基金Supported by the National Natural Science Foundation of China(11271105)the Key Research Project of Educational Department of Hubei Province(D20122202)Youth Research Project of Educational Department of Hubei Province(B20122203)
文摘In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to AT,S^(2).
基金This work was supported by the National Natural Science Foundation of China(Grant No.11771076)sponsored by Shanghai Sailing Program(Grant No.20YF1433100).
文摘We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations.This relation becomes a pre-order.Then,one-sided preorders associated to the weighted core-EP inverse are given from two perspectives.Finally,we make a comparison for these two sets of one-sided weighted pre-orders.
基金supported by the National Natural Science Foundation of China(Grant Nos.11771076,11871145)the Fundamental Research Funds for the Central Universitiesthe Postgraduate Research&:Practice Innovation Program of Jiangsu Province(No.KYCX19-0055).
文摘We first prove that if a is both left(6,c)-invertible and left(c,b)-invertible,then a is both(6,c)-invertible and(c,6)-invertible in a*-monoid,which generalizes the recent result about the inverse along an element by L.Wang and D.Mosic[Linear Multilinear Algebra,Doi.org/10.1080/03081087.2019.1679073],under the conditions(a6)*=ab and(ac)*=ac.In addition,we consider that ba is(c,fe)-invertible,and at the same time ca is(6,c)-invertible under the same conditions,which extend the related results about Moore-Penrose inverses studied by J.Chen,H.Zou,H.Zhu,and P.Patricio[Mediterr J.Math.,2017,14:208]to(6,c)-inverses.As applications,we obtain that under condition(a2)^(*)=a^(2),a is an EP element if and only if a is one-sided core invertible,if and only if a is group invertible.
基金supported by the National Natural Science Foundation of China(No.11771076,11961076)the Ministry of Science,Technology and Development,Republic of Serbia(No.174007)+1 种基金the China Postdoctoral Science Foundation(No.2020M671281)the Research Project of Hubei Provincial Departmentof Education(No.B2019128).
文摘In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of 1−pq,1−pqp,p−pq and p−q,where p,q are projections in different settings,such as∗-rings,∗-reducing rings and C∗-algebras.Moreover,several representations for the core inverses of product,difference and sum of two generalized projections are derived.In particular,a number of examples are given to illustrate our results.
基金The research was supported by the Ministry of Education,Science and Technological Development(grant no.174007)Republic of Serbia,and by NSF of China(11901510)+1 种基金NSF of Jiangsu Province of China(BK20170589)China Postdoctoral Science Foundation Funded Project(2017M611920).
文摘In this paper,we mainly give characterizations of EP elements in terms of equations.In addition,a related notion named a central EP element is defined and investigated.Finally,we focus on characterizations of a generalized EP element,i.e.,+-DMP element.