Representations of the (b, c)-inverses in Rings with Involution

Transcription

1 Filomat 31:9 (217), htts://doiorg/12298/fil179867k Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt://wwwmfniacrs/filomat Reresentations of the (b, c)-inverses in Rings with Involution Yuanyuan Ke a, Yuefeng Gao a, Jianlong Chen a a Deartment of Mathematics, Southeast University, Nanjing 2196, China Abstract Let R be a ring and b, c R The concet of (b, c)-inverses was introduced by Drazin in 212 In this aer, the existence and the exression of the (b, c)-inverse in a ring with an involution are investigated A new reresentation of the (b, c)-inverse based on the grou inverse is also resented 1 Introduction Throughout this aer, R denotes a ring with identity The set of all idemotents in R is denoted by R An involution of R is any ma : R R satisfying (a ) a, (ab) b a, (a + b) a + b for any a, b R An element a R is self-adjoint if a a An element R is a rojection if it is self-adjoint idemotent Let R, the range rojection of is a rojection R such that and 14 An element a of R is said to be (von Neumann) regular if there exists an element a of R such that aa a a a is called a {1}-inverse of a A solution to xax x is called an outer inverse of a An element a + of R is a {1, 2}-inverse of a if aa + a a and a + aa + a + hold In 11 a secial outer inverse, called (b, c)-inverse (see Definition 21), was introduced in the context of semigrous It is shown that the Moore-Penrose inverse (2), the Drazin inverse (1), the Chiman s weighted inverse (3, , or see 1, ), the Bott-Duffin inverse (2), the inverse along an element (15), the core inverse and dual core inverse (21) are all secial cases of (b, c)-inverses The urose of this article is to give necessary and sufficient conditions for the existence of the (b, c)- inverses in a ring with an involution, and derive new exressions for them, and then state some new roerties for these inverses Let a R and, R An element b R is the (, )-outer generalized inverse of a if bab b, ba, 1 ab (1) If the (, )-outer generalized inverse b exists, it is uniue 8 and denoted a (2), For more details about the (, )-outer generalized inverse see 4, 5, 7, 9, 12, 13, 17, Mathematics Subject Classification Primary 15A9 ; Secondary 16S5, 16U99, 2M99 Keywords Generalized inverse, (b, c)-inverse, (, )-outer generalized inverse, grou inverse Received: 2 November 215; Acceted: 9 February 216 Communicated by Dragana S Cvetković - Ilić Research suorted by the Foundation of Graduate Innovation Program of Jiangsu Province (No KYLX 8), Jiangsu Planned Projects for Postdoctoral Research Funds(No 15148B), the NSFC (No ), the Secialized Research Fund for the Doctoral Program of Higher Education (No ), the Natural Science Foundation of Jiangsu Province (No BK ) Corresonding author: Jianlong Chen addresses: keyy86@126com (Yuanyuan Ke), yfgao91@163com (Yuefeng Gao), jlchen@seueducn (Jianlong Chen)

2 If, R, then arbitrary x R can be written as or in the matrix form YY Ke et al / Filomat 31:9 (217), x x + x() + (1 )x + (1 )x(), x x11 x 12 x 21 x 22 where x 11 x, x 12 x(), x 21 (1 )x, x 22 (1 )x() If x (x ij ) and y (y ij ), then x + y (x ij + y ij ) Moreover, if r R and z (z ij ) r, then one can use usual matrix rules in order to multily x and z In a ring R with an involution, notice that x x x x x Recall that an element a R is grou invertible if there exists an uniue element a # R such that, aa # a a, a # aa # a #, aa # a # a We use R # to denote the set of all the grou invertible element of R If a R #, then a π 1 aa # is the sectral idemotent of a The following lemma was roved for Banach algebra elements in 16, it is correct in the ring case by elementary comutations Lemma 11 18, Lemma 14 a b (i) Let x R If a (R) s #, s ((1 )R(1 )) # and a π bs π, then a (ii) Let x c s x # a # (a # ) 2 bs π a # bs # + a π b(s # ) 2 s # R If a (R) #, s ((1 )R(1 )) # and s π ca π, then x # a # s π c(a # ) 2 s # ca # + (s # ) 2 ca π s # 2 The Reresentations of (b, c)-inverses in Rings with Involution In this section, we first recall the definition of the (b, c)-inverse and give some lemmas, and then investigate the reresentations of this inverse To discuss these matters more formally, we recall the definition of (b, c)-inverse in 11 Definition 21 11, Definition 13 Let R be any ring and let a, b, c R An element y R satisfying y (bry) (yrc), yab b and cay c (2) is called a (b, c)-inverse of a The element a of R has at most one (b, c)-inverse in R, and if the (b, c)-inverse y of a exists, it always satisfies yay y We denote by a (b,c) the (b, c)-inverse of a The (b, c)-inverse can reduces to classical inverse, Drazin inverse, Moore-Penrose inverse, the Bott-Duffin inverse, the inverse along an element, core inverse and dual core inverse denoted by a (1,1), a (aj,a j), a (a,a ), a (e,e), a (d,d), a (a,a ), a (a,a) resectively The following result is easy to check by the definition of the (b, c)-inverse

3 YY Ke et al / Filomat 31:9 (217), Lemma 22 Let R be any ring and let a, b, c R If a has a (b, c)-inverse, then b and c are both regular Proof If a has a (b, c)-inverse, using Definition 21, there is y R such that (2) holds This means there exist s, t R such that bsy y ytc, yab b, cay c Therefore, b yab bsyab bsb, c cay caytc ctc, that is, b and c are both regular Lemma 23 14, Theorem 21 Let R be a ring with an involution and R Then the following are euivalent: (i) + 1 is invertible in R; (ii) and ( ) exist The range rojections are uniue, given by the formula ( + 1) 1, ( ) ( + 1) 1 If a ring R with an involution has the GN-roerty (1 + xx R 1 for all x R), then every idemotent has a uniue range rojection Lemma 24 6, Lemma 22 Let R such that exists If f 1 +, then f R 1 and f 1 f 1 Base on the above facts, we have the following theorem Theorem 25 If b, c R are regular such that (bb ) and (c c) exist Let bb, c c, f 1 + and f 1 + For a R, then (i) a (b, c) exists if and only if a f 1 a1 a 2 a 3 a 4 and (a 3 ) (2),1 exists a (b, c) f 1,1 (1 ) f (3) (ii) a (b, 1 c c) exists if and only if a is reresented as in (3) and (a 1 ) (2) exists, a (b, 1 c c) f 1 (a1 ) (2), f (4) (iii) a (1 bb, c) exists if and only if a is reresented as in (3) and (a 4 ) (2) exists 1,1 a (1 bb, c) f 1 (a 4 ) (2) f 1,1 f (iv) a (1 bb, 1 c c) exists if and only if a is reresented as in (3) and (a 2 ) (2) exists 1, a (1 bb, 1 c c) f 1 (a 2 ) (2) 1, f

4 YY Ke et al / Filomat 31:9 (217), Proof We only give the roof of item (i), the rest are left to the reader by using similar techniues (i) Necessity Suose that a (b,c) exists, by Definition 21, there exists y R such that (2) holds This means y bsy ytc, yab b, cay c for some s, t R As b and c are regular and bb, c c, then y bb bsy bsy y, yabb bb, c cay c c, y yc c ytcc c ytc y, that is, y y, ya, ay, y() (5) As bb, c c are idemotents, we have the following reresentations of and : 1, 1 (6) (1 ) (1 ) Assume that a a 1 a 2 + a 1 1 a 3 1 a 1 a 4 + a 3 1 a 2 1 a 1 1 (1 ), where a 1 ( )R, a 2 ( )R(1 ), a 3 R, a 4 R(1 ) And suose a (b,c) b1 b y 2 b 3 b 4 From (5), y y gives b 3 b 4 Since y(), we obtain b 1 b 2 1 The euality ya imlies b 2 a 3, b 2 a 3 1 1, that is b 2 a 3 Similarly, ay can reduce to a 3 b 2 Note that yay y imlies b 2 a 3 b 2 b 2 By (1), we get at once b 2 (a 3 ) (2),(1 ), and Furthermore, and f 1 f 1 a (b,c) a1 a 2 f a 3 a 4 (1 ),1 f,(1 ) 1 (a 3 ) (2),(1 ) a, 1 (1 ) (1 ) a1 a 2 a 3 a 4 (1 ) a 1 a 2 + a 1 1 a 3 1 a 1 a 4 + a 3 1 a 2 1 a ,1,(1 ) 1 (a 3 ) (2),(1 ) a (b,c) Sufficiency If a has the form (3), let y f 1 y 1 1,1,(1 ) 1 (a 3 ) (2),(1 ) 1 1 (1 ),1 f, then 1 1 (1 ) (1 ) (1 ) (1 )

5 YY Ke et al / Filomat 31:9 (217), So we have bb y y yc c y yabb ya f 1 1,(1 ) 1 (a 3 ) (2),(1 ),(1 ) 1 (a 3 ) (2),(1 ) y,,(1 ) 1 (a 3 ) (2),(1 ),(1 ) 1 (a 3 ) (2),(1 ),1 1 bb, a1 a 2 a 3 a 4 1 y, f (1 ) (1 ) (1 ) 1 c cay ay 1 1 (1 ) (1 ) (1 ) (1 ) f 1 a1 a 2 a 3 a 4 c c (1 ) Therefore, we have y bry yrc, and yab b, cay c, that is, a (b,c) exists,1 f Theorem 26 If b, c R are regular such that (bb ) and (c c) exist Let bb, c c, f 1 + and f 1 + For d R, then (i) d (1 c c, 1 bb ) exists if and only if d f 1 d1 d 2 d 3 d 4 and (d 3 ) (2) 1, exists d (1 c c, 1 bb ) f 1 (d3 ) (2) 1, f (7) (1 ) f (8) (ii) d (1 c c, bb ) exists if and only if d is reresented as in (7) and (d 1 ) (2) exists 1,1 d (1 c c, bb ) f 1 (d1 ) (2) 1,1 f (1 ) (iii) d (c c, 1 bb ) exists if and only if d is reresented as in (7) and (d 4 ) (2) exists, d (c c, 1 bb ) f 1 (d 4 ) (2) f, (1 )

6 YY Ke et al / Filomat 31:9 (217), (iv) d (c c, bb ) exists if and only if d is reresented as in (7) and (d 2 ) (2) exists,1 d (c c, bb ) f 1 (d 2 ) (2),1 (1 ) f Proof We only give the roof of item (i), the rest are left to the reader by using similar techniues (i) Necessity Assume that d (1 c c, 1 bb ) exists, according to Definition 21, there is y R such that y (1 c c)ry yr(1 bb ), yd(1 c c) 1 c c, (1 bb )dy 1 bb So there exist s, t R such that y (1 c c)sy yt(1 bb ), yd(1 c c) 1 c c, (1 bb )dy 1 bb Since bb, c c, we know (1 )y (1 c c)(1 c c)sy (1 c c)sy y, y ybb yt(1 bb )bb So we have y ()y, y, yd(), (1 )dy 1 Let and be reresented as in (6) Denote by d d1 + d d 3 1 d 4 1 d 2 1 d 4 d 3 + d 4 1 d 4, where d 1 R( ), d 2 R, d 3 (1 )R( ), d 4 (1 )R And suose d (1 c c,1 bb ) y b1 b 2 b 3 b 4 (1 ) From y, we have b 1 As y (1 )y, we get b 3 1 b 1, b 4 1 b 2 The condition yd(1 ) 1 imlies b 2 d 3, 1 b 2 d 3 1 The euality (1 )dy 1 gives 1 d 3 b 2 1, d 3 b 2 1 Notice that ydy y, which imlies b 2 d 3 b 2 b 2 By (1), we can deduce that b 2 (d 3 ) (2) 1, So we have d (1 c c,1 bb ) (d 3 ) (2) 1 (d 3 ) (2) 1, 1, (1 ) Therefore, f 1 d1 d 2 d 3 d 4 f d1 + d d 3 1 d 4 1 d 2 1 d 4 d 3 + d 4 1 d 4 d And f 1 (d3 ) (2) 1, f (1 ) Sufficiency If d has the form (7), let y f 1 y 1 (d 3 ) (2) 1 (d 3 ) (2) (1 ) (1 ) 1, 1, (1 ) (d 3 ) (2) 1 (d 3 ) (2) 1, 1, (1 ) d (1 c c,1 bb ) (d3 ) (2) 1, f, we have (1 ) (d3 ) (2) 1, 1 1 (1 )

7 So we have (1 c c)y YY Ke et al / Filomat 31:9 (217), y(1 bb ) yd(1 c c) f 1 (1 bb )dy 1 (d 3 ) (2) 1 (d 3 ) (2) (1 ) (1 ) 1, 1, (d 3 ) (2) 1 (d 3 ) (2) (d 3 ) (2) 1 (d 3 ) (2) (d3 ) (2) 1, 1 (1 ) (1 ) 1, 1, 1, (1 ) 1, (d 3 ) (2) 1 (d 3 ) (2) (1 ) y, (1 ) (1 ) y, d1 d 2 d 3 d 4 1 c c, 1 1 f 1 d1 d 2 d 3 d bb 1, 1, (1 ) 1 1 f 1 (1 ) (1 ) (d3 ) (2) 1, f (1 ) Therefore, we have y (1 c c)ry yr(1 bb ), yd(1 c c) 1 c c, (1 bb )dy 1 bb, that is, d (1 c c,1 bb ) exists The next theorem gives some roerties of the (b, c)-inverse Theorem 27 Let a, b, c R Then (i) a (b,c) exists if and only if (a ) (c,b ) exists In addition, (a (b,c) ) (a ) (c,b ) (ii) If a (b,c) exists, then (a (b,c) ) 2 a (b,c) if and only if a (b,c) b b Proof (i) By Definition 21, a (b,c) exists if and only if there exists y R such that which is euivalent to there is y R such that y bry yrc, yab b, cay c, y c Ry y Rb, y a c c, b a y b That is, (a ) (c,b ) exists, and (a (b,c) ) (a ) (c,b ) (ii) If a (b,c) exists and (a (b,c) ) 2 a (b,c), it follows that b a (b,c) ab (a (b,c) ) 2 ab a (b,c) (a (b,c) ab) a (b,c) b Conversely, if a (b,c) exists and a (b,c) b b, from Definition 21, there is s R such that a (b,c) bsa (b,c) a (b,c) bsa (b,c) (a (b,c) ) 2 In the following result, we consider b c e R Theorem 28 Let a R and e R such that e exists Let f e 1 + e e If a (e,e) exists, then

8 (i) a (e,e) a aa (e,e) if and only if a fe 1 a1 a 4 YY Ke et al / Filomat 31:9 (217), e e f e (9) (ii) a (e,e) is self-adjoint if and only if e and e ae are self-adjoint Proof The roof is left to the reader since it is same as the roof of 18, Theorem 25 Now we will resent a reresentation of the (b, c)-inverse based on grou inverse Theorem 29 If b, c R are regular such that (bb ) and (c c) exist Let bb, c c, f 1 + and f 1 + Suose a, d R such that a (bb, 1 c c) and d (1 c c, bb ) exist Then (i) d()a, d()a R #, (ii) ad(), ()ad() R #, (iii) d()a, ()ad() R #, a (bb, 1 c c) d()a # d() d()a # d(), d (1 c c, bb ) ()ad()a # a (bb, 1 c c) d()ad() # d()ad() #, d (1 c c, bb ) ()ad() # ()a a (bb, 1 c c) d()()ad() #, d (1 c c, bb ) ()ad()a # Proof The roof is left to the reader since it is same as the roof of 18, Theorem 26 Let R be a ring with an involution and has the GN-roerty (that is, for all x R, 1 + x x is invertible), then we can omit the assumtion and exist in the revious results Let A be a C -algebra with unit 1 An element x of A is ositive (denoted by x ) if x x and σ(x), + ), where σ(x) denotes the sectrum of x Theorem 21 Let a, d A If b, c A are regular such that (bb ) and (c c) exist Take bb and c c Suose a (bb, 1 c c) exists and a (bb, 1 c c), then (a + dd ) (bb, 1 c c) exists and (a + dd ) (bb, 1 c c) a (bb, 1 c c) a (bb, 1 c c) d(1 + d a (bb, 1 c c) d) 1 d a (bb, 1 c c) Proof The condition a (bb, 1 c c) gives a (bb, 1 c c) s s, for some s A So we have d a (bb, 1 c c) d d s sd (sd) sd, which imlies that 1 + d a (bb, 1 c c) d is invertible Denote by x a + dd and y a (bb, 1 c c) a (bb, 1 c c) d(1+d a (bb, 1 c c) d) 1 d a (bb, 1 c c) Then, it is easy to get bb y y, y(1 c c) y since bb a (bb, 1 c c) a (bb, 1 c c) and a (bb, 1 c c) (1 c c) a (bb, 1 c c) Notice that a (bb, 1 c c) abb bb and (1 c c)aa (bb, 1 c c) 1 c c, so we have yxbb a (bb, 1 c c) abb + a (bb, 1 c c) dd bb a (bb, 1 c c) d(1 + d a (bb, 1 c c) d) 1 (d a (bb, 1 c c) abb + d a (bb, 1 c c) dd bb ) bb + a (bb, 1 c c) dd bb a (bb, 1 c c) d(1 + d a (bb, 1 c c) d) 1 (1 + d a (bb, 1 c c) d)d bb bb,

9 and YY Ke et al / Filomat 31:9 (217), (1 c c)xy (1 c c)aa (bb, 1 c c) + (1 c c)dd a (bb, 1 c c) (1 c c) (aa (bb, 1 c c) d + dd a (bb, 1 c c) d)(1 + d a (bb, 1 c c) d) 1 d a (bb, 1 c c) 1 c c + (1 c c)dd a (bb, 1 c c) (1 c c)d(1 + d a (bb, 1 c c) d) (1 + d a (bb, 1 c c) d) 1 d a (bb, 1 c c) 1 c c Thus, we obtain y bb Ay ya(1 c c) and yxbb bb, (1 c c)xy 1 c c, so we can conclude that x (bb, 1 c c) exists and x (bb, 1 c c) y References 1 A Ben-Israel, T N E Greville, Generalized Inverses: Theory and Alications, second ed, Sringer, NY, 23 2 R Bott, R J Duffin, On the algebra of networks, Trans Amer Math Soc 74 (1953) T L Boullion, P L Odell (Eds), Proceedings of the Symosium on Theory and Alications of Generalized Inverses, Texas Tech Press, Lubbock, J Cao, Y Xue, The characterizations and reresentations for the generalized inverses with rescribed idemotents in Banach algebras, Filomat 27 (5) (213) D S Cvetković-Ilić, X Liu, J Zhong, On the (, )-outer generalized inverse in Banach Algebras, Al Math Comut 29 (29) C Deng, Y Wei, Characterizations and reresentations of the (P, Q)-outer generalized inverse, Al Math Comut 269 (215) D S Djordjević, P S Stanimirović, Y Wei, The reresentation and aroximations of outer generalized inverses, Acta Math Hungar 14 (24) D S Djordjević, Y Wei, Outer generalized inverses in rings, Comm Algebras 33 (25) D S Djordjević, Y Wei, Oerators with eual rojections related to their generalized inverses, Al Math Comut 155 (24) M P Drazin, Pseudo-inverses in associative rings and semigrous, Amer Math Monthly 65 (1958) M P Drazin, A class of outer generalized inverses, Linear Algebra Al 436 (212) G Kantún-Montiel, Outer generalized inverses with rescribed ideals, Linear Multilinear Algebra 62 (9) (214) M Z Kolundžija, (, )-outer generalized inverse of block matrices in Banach algebras, Banach J Math Anal 8 (1) (214) J J Koliha, V Rakocević, Range rojections and the Moore-Penrose inverse in rings with involution, Linear Multilinear Algebra 55 (27) X Mary, On generalized inverses and Green s relations, Linear Algebra Al, 434 (8) (211) D Mosić, D S Djordjević, Rerezentation for the generalized Drazin inverse of block matrices in Banach algebras, Al Math Comut 218 (212) D Mosić, D S Djordjević, G Kantún-Montiel, Image-kernel (P, Q)-inverses in rings, Electron J Linear Algebra 27 (214) D Mosić, Characterizations of the image-kernel (, )-inverses, Bull Malays Math Sci Soc (2) (acceted) 19 B Načevska, D S Djordjević, Outer generalized inverses in rings and related idemotents, Pul Math Debrecen 73 (3-4) (28) R Penrose, A generalized inverse for matrices, Math Proc Cambridge Philos Soc 51 (1955) D S Rakić, N Č Dinčić, D S Djordjević, Grou, Moore-Penrose, core and dual core inverse in rings with involution Linear Algebra Al 463 (214)