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1、精品论文http:/Additive maps seemingly preserving group inverses Jian-Ying Ronga, Xu-Qing Liub a. Department of Foundation Courses, Huaian College of Information Technology, Huaian 223003, PR China;b. Department of Computing Science, Huaiyin Institute of Technology, Huaian 223001, PR ChinaAbstract: Addit
2、ive preserver problems on algebra structures over square matrices spaces are of considerable interest to many authors in the past decade, in which the notion additive map preserving group inverses is mentioned. In the present note, we first propose a suitably acceptable notion additive map seemingly
3、 preserving group inverses of matrices over fields with characteristic not 2 or 3 and then investigate the general characterization of this notion as a beneficial attempt.Keywords: Additive map, Preserving group inverses, Seemingly preserving group inversesAMS(2000) Classification: 15A04, 15A331 Int
4、roductionLet F be a field of characteristic not 2 or 3. Denote by F the multiplicative group, Mn (F) the algebra of all n n matrices over F, GLn (F) the general linear group of Mn (F), Ei j the n n matrix with 1 in the (i, j) position and 0 elsewhere.For given A Mn (F), if there exists (and therefor
5、e must be unique) B Mn (F) satisfies(1) ABT A = A, (2) BT ABT = BT , (3) ABT = BT A,we call BT the group inverse of A, and write BT = A# , where BT is the transpose of B. For the sake of convenience, we writeDi j = Ei j + E ji for i, j, i , j throughout this note.In the past decade, linear and addit
6、ive preserver problems are quite active and of considerable interest to some authors; e.g., cf. 3,4,5 and 1,2,6,7,8, respectively. A notion of additive map preserving group inverses, restated in a definition version in the following Definition 1, was considered in 1, also see among others. In this n
7、ote, we suggest to consider another notion of additive map seemingly preserving group inverses stated in the following Definition 2 and try investigating its description.Definition 1.1 We call an additive map f : Mn (F) Mn (F) preserving group inverses of matrices, if f (BT ) = f (A)# for any BT = A
8、# with A, B Mn (F). Definition 1.2 We call an additive map f : Mn (F) Mn (F) seemingly preserving group inverses of matrices, if f (B)T =f (A)# for any BT = A# with A, B Mn (F). We denote by F the set of all additive maps seemingly preserving group inverses of matrices. In the following section, we
9、will give the characterization of F .2 Additive map seemingly preserving group inversesNote that throughout this note any f F is assumed to be additive. We first give several lemmas.Lemma 2.1 Assume f F . Then for any a, b F and mutually distinct i, j, k 1, n, the statements below hold, where1, n =
10、1, 2, , n (the same below), namely,(1) f (aEii )T = f (a1 Eii )# ;(2) f (a1 Eii b1 E j j )T = f (aEii bE j j )# ; This research was supported by Grants HGQ0637 and HGQN0725 and the “Green & Blue Project” Program for 2008 to Cultivate Young CoreInstructors from Huaiyin Institute of Technology, Jiangs
11、u Province, P.R. China. Corresponding author. Email address: (J. Rong); (X. Liu).3精品论文(3) f (In a E ji )T = f (In aEi j )# ;(4) f (Ei j E ji )T = f (Ei j E ji )# ;(5) f ( E j j + aE ji + bE jk )T = f (E j j + aEi j + b Ek j )# ;2(6) 1 f (a Eii aE j j aDi j )T = f (a1 Eii a1 E j j a1 Di j )# ;2(7)
12、f (Eii + 1 E j j E ji )T = f (Eii + 2E j j 2Ei j )# ; (8) f (Eii + 2E j j Di j )T = f (2 Eii + E j j Di j )# ; (9) f (a2 bEii aDi j )T = f (bE j j a1 Di j )# ;(10) f (Eii + E j j D ji )T = f (Eii + E j j Di j )# ; (11) f (Eii E j j aE ji )T = f (Eii E j j aEi j )# ; (12) f (Di j )T = f (Di j )# ;(13
13、) f (Di j Ekk )T = f (Di j Ekk )# . Lemma 2.2 Assume f F . Then f (aEii ) f (bE j j )T = f (bE j j ) f (aEii ) = 0 for any i, j, i , j, and a, b F.Proof. From (2) of Lemma 1 and f F , we havef (aEii bE j j ) = f (aEii bE j j ) f (a1 Eii b1 E j j )T f (aEii bE j j ).By the above two equations adding
14、and the characteristic of F not 2, we obtainf (aEii ) f (b1 E j j )T f (bE j j ) + f (bE j j ) f (b1 E j j )T f (aEii ) + f (bE j j ) f (a1 Eii )T f (bE j j ) = 0.By the characteristic of F not 2 or 3, a F arbitrary and f additive, we getf (bE j j ) f (a1 Eii )T (bE j j ) = 0,(2.1)f (aEii ) f (b1 E
15、j j )T (bE j j ) + f (bE j j ) f (b1 E j j )T f (aEii ) = 0.(2.2) Post-multiplying (2.2) by f (b1 E j j )T and noting that (2.1) holds,0 = f (aEii ) f (b1 E j j )T (bE j j ) f (b1 E j j )T = f (aEii ) f (b1 E j j )Tcombining (1) of Lemma 1. Similarly, f (b1 E j j )T f (aEii ) = 0. Lemma 2.3 Assume f
16、 F . Then for any i, j, i , j, and a F,f (aEi j ) = f (aEi j ) f (Eii )T f (Eii ) + f (Eii ) f (Eii )T f (aEi j ) = f (aEi j ) f (E j j )T f (E j j ) + f (E j j ) f (E j j )T f (aEi j ),and f (aEi j ) f (a E ji )T f (aEi j ) = f (aEii ) f (aEi j )T f (aEii ) = f (aE j j ) f (aEi j )T f (aE j j ) = 0
17、.Proof. From (3) of Lemma 1 and f F , we get f (In aEi j ) = f (In aEi j ) f (In aE ji )T f (In aEi j ). By the above two equations adding, a F arbitrary, f additive, and the characteristic of F not 2, f (aEi j ) f (aE ji )T f (aEi j ) = 0 andf (aEi j ) = f (aEi j ) f (In )T f (In ) + f (In ) f (In
18、)T f (aEi j ) f (In ) f (aE ji )T f (In ).Pre-multiplying and post-multiplying the last equation by f (E j j )T and combining Lemma 2 and (1) of Lemma 1, we havef ( E j j )T f (aEi j ) f (E j j )T f ( E j j )T f (E j j ) f (aEi j )T f (E j j ) f ( E j j )T = 0(2.3) Taking b = 0 in (5) of Lemma 1, we
19、 obtainf (aEi j ) = f (aEi j ) f (Eii )T f (Eii ) + f (Eii ) f (Eii )T f (a Ei j ) + f (Eii ) f (aEi j )T f (Eii )= f (aEi j ) f (E j j )T f (E j j ) + f (E j j ) f (E j j )T f (aEi j ) + f (E j j ) f (a Ei j )T f (E j j ).Pre-multiplying and post-multiplying the last equation by f (E j j )T , we ha
20、vef (E j j )T f (aEi j ) f (E j j )T + f (E j j )T f (E j j ) f (aEi j )T f (E j j ) f ( E j j )T = 0,which combining equation (2.3) yields f (E j j ) f (aEi j )T f (E j j ) = 0. Similarly, we obtainf (Eii ) f (aEi j )T f (Eii ) = 0.Consequently, f (aEi j ) = f (aEi j ) f (Eii )T f (Eii ) + f ( Eii
21、) f (Eii )T f (aEi j ) = f (aEi j ) f (E j j )T f (E j j ) + f (E j j ) f (E j j )T f (aEi j ). Lemma 2.4 Assume f F . Then the following are equivalent:(i) f = 0;(ii) f (Di j ) = 0 for some i , j; (iii) f (Eii ) = 0 for some i. Proof.(i)(ii): It is clear.(ii)(iii): If f (Di j ) = 0, for some i, j,
22、i , j, from (6) and (2) of Lemma 1, we have31212f (Eii E j j )T =f (Eii E j j Di j )T = f (Eii E j j Di j )# = f (Eii E j j )# = f (Eii E j j )T ,hence f (Eii ) = f (E j j ). By (1) and (9) of Lemma 1,f (Eii )T = f (Eii + Di j )T = f ( E j j Di j )# = f (E j j )# = f (E j j )T = f (Eii )T ,which imp
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