Representation for the generalized Drazin inverse of block matrices in Banach algebras Dijana Mosić and Dragan S. Djordjević Abstract Several representations of the generalized Drazin inverse of a block matrix with a group invertible generalized Schur complement in Banach algebra are presented. Key words and phrases: generalized Drazin inverse, Schur complement, block matrix. 2010 Mathematics subject classification: 46H05, 47A05, 15A09. 1 Introduction The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, iterative methods in numerical linear algebra, etc. Representations for the Drazin inverse of block matrices under certain conditions where given in the literature 1, 3, 4, 5, 6, 7, 9, 14, 17. In this paper, we present formulas for the generalized Drazin inverse of block matrix with generalized Schur complement being group invertible in Banach algebra. Moreover, necessary and sufficient conditions for the existence as well as the expressions for the group inverse of triangular matrices are obtained. Let A be a complex unital Banach algebra with unit 1. For a A, we use σa) and ρa), respectively, to denote the spectrum and the resolvent set of a. The sets of all nilpotent and quasinilpotent elements σa) {0}) of A will be denoted by A nil and A qnil, respectively. The generalized Drazin inverse of a A or Koliha Drazin inverse of a) is the element b A which satisfies bab b, ab ba, a a 2 b A qnil. The authors are supported by the Ministry of Education and Science, Republic of Serbia, grant no. 174007. 1
If the generalized Drazin inverse of a exists, it is unique and denoted by a d, and a is generalized Drazin invertible see 11). Recall that p 1 aa d is the spectral idempotent of a corresponding to the set {0}, and it will be denoted by a π. The set of all generalized Drazin invertible elements of A is denoted by A d. The Drazin inverse is a special case of the generalized Drazin inverse for which a a 2 b A nil instead of a a 2 b A qnil. Obviously, if a is Drazin invertible, then it is generalized Drazin invertible. The group inverse is the Drazin inverse for which the condition a a 2 b A nil is replaced with a aba. We use a to denote the group inverse of a, and we use A to denote the set of all group invertible elements of A. Let p p 2 A be an idempotent. Then we can represent element a A as a11 a a 12, a 21 a 22 where a 11 pap, a 12 pa1 p), a 21 1 p)ap, a 22 1 p)a1 p). We use the following lemmas. Lemma 1.1. Let b A d and a A qnil. i) 2, Corollary 3.4 If ab 0, then ab A d and ab) d b d ) n1 a n. ii) If ba 0, then a b A d and a b) d a n b d ) n1. Specializing 2, Corollary 3.4 with multiplication reversed) to bounded linear operators Castro González et al. 2 recovered 8, Theorem 2.2 which is a spacial case of Lemma 1.1ii). By the following lemma, Castro González et al. 2 recovered 10, Theorem 2.1 for matrices and 8, Theorem 2.3 for bounded linear operators. Lemma 1.2. 2, Example 4.5 Let a, b A d and let ab 0. Then a b) d b d ) n1 a n a π b π b n a d ) n1. The following result is well-known for complex matrices see 16) and it is proved for elements of Banach algebra in 12. a b Lemma 1.3. 12, Lemma 2.2 Let x A relative to the idempotent p A, a pap) d and let w aa d a d bca d be such that aw pap) d c d. 2
If ca π 0, a π b 0 and the generalized Schur complement s d ca d b is equal to 0, then x d p 0 aw) d 2 a 0 p a d b ca d. 1) 0 Let a b x c d A 2) relative to the idempotent p A, a pap) d and let the generalized Schur complement s d ca d b 1 p)a1 p)). The generalized Schur complement s plays an important role in the representations for x d in many cases 9, 14, 15, 17. Hartwig et al. 9 presented representations for the Drazin inverse of a 2 2 block matix under conditions which involve W AA D A D BCA D and that the generalized Schur complement is equal to 0. Li 13 investigated a representation for the Drazin inverse of block matrices with a singular and group invertible generalized Schur complement, recovering the formula 1) for complex matrices 16. We investigate representations of the generalized Drazin inverse of a block matrix x in 2) with a group invertible generalized Schur complement s d ca d b under different conditions. Thus, we extend some results from 13, 16 to more general settings. Also, we obtain equivalent condition for the existence and representations for the group inverse of triangular matrices in Banach algebra. The paper s aim is to further weaken the conditions on the elements needed to produce explicit formulae for the generalized Drazin inverse of x compared to those known from the literature. Such formulae are very complicated, but the main goal is to establish that x has the generalized Drazin inverse, and the formulae are the means to produce that result. 2 Results In this section, we assume that i) the element x is defined as in 2) relative to the idempotent p A, ii) a pap) d, iii) s d ca d b and s 1 p)a1 p)), 3
iv) w aa d a d bs π ca d. First, we give a formula for the generalized Drazin inverse of block matrix x in 2) in terms of w aa d a d bs π ca d and the group invertible Schur complement s. Theorem 2.1. If aw pap) d, a π b 0, bs π ca π 0, wbss 0, ss ca d bss 0, 3) then x A d and ) x d 0 bs 1 0 ca d bs s cwaw) d 2 a s s cwaw) d 2 bs π ) s ) n1 ca n 1 a π v n a v n bs π n1 p bs 0 1 p) ds r, 4) where v n s ) n1 ca d aw) n 1 aw) π, n 1, 2,... ), and aw) r d 2 a aw) d 2 bs π ca d aw) d 2 a ca d aw) d 2 bs π. Proof. Using the equalities aa d a π p, ss s π 1 p and ss π 0, note that a x 2 a d bs π aa π bss caa d ca d bs π ca π dss : y z. By a d a π 0 and 3), we obtain bs yz π ca π awbss ca d bs π ca π cwbss 0. To check that y A d, set A y a 2 a d, B y bs π, C y caa d and D y ca d bs π. Since a 2 a d ) a d, A y pap) and S y D y C y A y B y 0. Also, A π y B y a π bs π 0, C y A π y 0 and W y A y A y A y B y C y A y w. Applying Lemma 1.3, observe that y A d and y d p 0 ca d 0 aw) d 2 a 0 4 p a d bs π r.
In order to prove that z A d, we write z aa π bss ca π s 0 ca d bss : z 1 z 2 z 3. m t Recall that, for u, 0 n λ ρ pap m) ρ 1 p)a1 p) n) λ ρu), i.e. σu) σ pap m) σ 1 p)a1 p) n). From aa π pap) qnil and s A, we conclude that z 1 A qnil, z 2 A and z 2 0 s. Using Lemma 1.1i), by z 1 z 2 0, we get z 1 z 2 A d and z 1 z 2 ) d z 2 )n1 z1 n. Further, z2 3 0, i.e. z 3 A nil and z 1 z 2 )z 3 0. By Lemma 1.1ii), z A d and z d z 1 z 2 ) d z 3 z 1 z 2 ) d 2. Therefore, by Lemma 1.2, we deduce that x A d and x d z d ) n1 y n y π z π z n y d ) n1 1 z 3 z 1 z 2 ) d )z 1 z 2 ) d n1 y n y π z π z n y d ) n1 : X 1 X 2, 5) where X 1 1 z 3 z 1 z 2 ) d )x 1 y π, x 1 z 1 z 2 ) d n1 y n, X 2 z π y d z π z n y d ) n1. n1 5
Since z 1 y 0, then z 1 z 2 ) d y 0 s x 1 z 1 z 2 ) d z 1 z 2 ) d z 1 z 2 ) d Now, we have X 1 1 1 z 1 z 2 ) d n z 2 yn n1 n1 z 2 k1 y z 2 y and ) n z 2 )k1 z1 k z 2 yn ) z 2 )n z 2 )n 1 z 2 )k1 z1 k n1 z 2 )n1 z1 n z 2 )n1 y n. 0 b 0 ca d b 0 b 0 ca d b k0 n1 k1 z 2 yn ) ) z 2 )k1 z1 k z 2 )n1 z1 n z 2 )n1 y n ) z 2 z 2 )k1 z1 k k1 ) z 2 )n z1 n z 2 )n y n 1 1 0 bs n1 0 ca d bs 0 bs 0 ca d bs y π ) z 2 yπ ) z 2 yπ n1 z 2 n1 ) z 2 )n1 z1 n y π z 2 )n1 y n y π n1 n1 y π ) z 2 )n1 z1 n z 2 )n1 y n y π.6) n1 Observe that aa d aw) aw aw)aa d, y π 1 yy d p aw) d a cwaw) d 2 a aw) d bs π 1 p) cwaw) d 2 bs π and y n y π aa d aw) n 1 aw) π a aw) n 1 aw) π bs π ca d aw) n 1 aw) π a ca d aw) n 1 aw) π bs π n 1, 2,... ). 6
Hence, these equalities and 6) give ) 0 bs X 1 1 0 ca d bs s cwaw) d 2 a s s cwaw) d 2 bs π a n a π 0 0 s ) n1 ca n 1 a π 0 n1 ) s ) n1 ca d aw) n 1 aw) π a s ) n1 ca d aw) n 1 aw) π bs π n1 ) 0 bs 1 0 ca d bs s cwaw) d 2 a s s cwaw) d 2 bs π ) s ) n1 ca n 1 a π v n a v n bs π. 7) From n1 zz d y z 1 z 2 )z 1 z 2 ) d z 3 z 1 z 2 ) d y z 1 z 2 )z 2 y z 3z 2 y z 2 z 3 )z 2 y, we get zz d y d z 2 z 3 )z p bs 2 yd and z π y d 0 1 p) ds r. Thus, by p bs z 0 1 p) ds z 1 and a π aw) d a π aa d awaw) d 2 0, notice p bs that zz π y d z 0 1 p) ds r z 1 r 0 and X 2 z π y d So, 5), 7) and 8) imply 4). z n z π y d ) n1 z π r. 8) n1 Our conditions a π b 0, bs π ca π 0, wbss 0, ss ca d bss 0 in Theorem 2.1 can be formulated geometrically as ba aa, s π ca π A b, bss A w, ca d bss A s, where e {f A : ef 0}. If we assume that a is a group invertible and w 1 in Theorem 2.1, we obtain the following result as a consequence. 7
Corollary 2.1. Let a pap) and let w aa a bca 1. If a π b 0, bca π 0 and bss 0, then x A d and x d a a ) 2 b s ca s ) 2 ca π s π ca ) 2 s s ca ) 2 b s π ca ) 3. b In the following theorem, the other formula for the generalized Drazin inverse of block matrix is presented. Theorem 2.2. If aw pap) d, ca π 0, a π bs π c 0, ss cw 0, ss ca d bss 0, then x A d and x d 0 aw) d bs 0 s s π ca d aw) d bs 0 a n 1 a π bs ) n1 aa d aw) n 1 aw) π bs ) n1 ) 0 s π ca d aw) n 1 aw) π bs ) n1 n1 ) p 0 1 s c s ca d t b s c 1 p) s, 9) d where t aw) d 2 a s π ca d aw) d 2 a aw) d 2 b s π ca d aw) d 2 b. Proof. If we write a x 2 a d aa d b s π c s π ca d b aa π a π b ss c ss d : y z, then zy 0. Using Lemma 1.3, we conclude that y A d and y d t. To show that z A d, let aa π a z π b 0 s ss c ss ca d : z b 1 z 2 z 3. Then z 1 A qnil, z 2 A and z 2 z 1 0. By Lemma 1.1ii), z 1 z 2 A d and z 1 z 2 ) d z1 nz 2 )n1. From z3 2 0, z 3z 1 z 2 ) 0 and Lemma 1.1i), z A d and z d z 1 z 2 ) d z 1 z 2 ) d 2 z 3. 8
Applying Lemma 1.2, observe that x A d and x d y d ) n1 z n z π y π y n z d ) n1 : X 1 X 2. 10) The equalities yz 1 0 and yz 1 z 2 ) d yz 2 imply y d z π p 0 t s c 1 p) s and y d d z π z 0. Next, we obtain X 2 X 1 y d z π y π y n z d ) n1 Since y π y π z 1 z 2 ) d y π y d ) n1 z π z n tz π. 11) n1 y π y n z 1 z 2 ) d n1 1 z 1 z 2 ) d z 3 ) ) y π y n z 2 z 1 z 2 ) d n 1 z 1 z 2 ) d z 3 ) n1 ) z1 n z 2 )n1 y π y n z 2 z 2 )n z1 k z 2 )kn 1 z 2 z 3 y π n1 ) z1 n z 2 )n1 z 3 n1 z n 1 z 2 )n1 n1 k1 y π y n z 2 )n1 1 z 2 z 3) n1 y π z 2 z1 n z 2 )n1 y π y n z 2 )n1 1 z 2 z 3). p aw) d a s π ca d aw) d a n1 aw) d b 1 p) s π ca d aw) d and b y n y π aa d aw) n 1 aw) π a aa d aw) n 1 aw) π b s π ca d aw) n 1 aw) π a s π ca d aw) n 1 aw) π b n 1, 2,... ), 9
we get X 2 0 aw) d bs 0 s s π ca d aw) d bs 0 a n 1 a π bs ) n1 aa d aw) n 1 aw) π bs ) n1 0 s π ca d aw) n 1 aw) π bs ) n1 n1 ) 1 s c s ca d b ) 12) Thus, from 10), 11) and 12), we obtain 9). Notice that explicit formulae 4) and 9) for the generalized Drazin inverse of x are complicated, but the conditions on the elements needed to produce these formulae are weaken than those known from the literature and x has the generalized Drazin inverse under these conditions. In Theorem 2.2, supposing that a pap) and w 1, the next corollary follows. Corollary 2.2. Let a pap) and let w aa a bca 1. If ca π 0, a π bc 0 and ss c 0, then x A d and x d a a bs a π bs ) 2 a ) 2 bs π s π ca ) 2 s ca ) 2 bs s π ca ) 3 bs π. The following result is a consequence of Theorem 2.1 and Theorem 2.2. Corollary 2.3. If aw pap) d, s 0 and if one of the following conditions holds: i) a π b 0 and ca π 0, ii) a π b 0 and bca π 0, iii) ca π 0 and a π bc 0, then x A d and x d aw) d 2 a ca d aw) d 2 a aw) d 2 b ca d aw) d 2 b. Notice that, the preceding corollary recover Lemma 1.3 for elements of Banach algebra and the analogy result for matrices 16. We will use Theorem 2.1 to find the group inverse of a triangular block matrix. Precisely, for b 0 in Theorem 2.1, we obtain the next result. 10
a 0 Theorem 2.3. Let x A relative to the idempotent p A, c s a pap) d and s 1 p)a1 p)). Then i) x A d and x d a d 0 s π ca d ) 2 s ca d s n1 s ) n1 ca n 1 a π 0 ; 13) ii) x A if and only if a pap) and s π ca π 0. Furthermore, if a pap) and s π ca π 0, then x a 0 s π ca ) 2 s ca s ) 2 ca π s Proof. i) If b 0 in Theorem 2.1, then s d, w aa d, aw a 2 a d pap), aw) a d and a d aw) π a d a π 0 implying 13). ii) By the part i), note that a 2 a d 0 x 2 x d caa d s 2 s ) n1 ca n 1 a π. s n1 Consequently, x 2 x d x a 2 a d a and caa d ss ) n ca n 1 a π c. n1 Hence, x A is equivalent to a pap) and ss ca π ca π. In the same manner as in the proof of Theorem 2.3, if c 0 in Theorem 2.2, we verify the following theorem in which necessary and sufficient condition for the existence and representation of the group inverse are considered. a b Theorem 2.4. Let x A relative to the idempotent p A, 0 s a pap) d and s 1 p)a1 p)). Then. i) x A d and x d a d a d ) 2 bs π a d bs 0 s 0 a n 1 a π bs ) n1 n1 ; 11
ii) x A if and only if a pap) and a π bs π 0. Furthermore, if a pap) and a π bs π 0, then x a a ) 2 bs π a bs a π bs ) 2 0 s. Observe that the part i) of Theorem 2.3 and the same part of Theorem 2.4 are the special cases of 2, Theorem 2.3 for Banach algebra elements and 8, Theorem 2.2 for bounded linear operators. Finally, we give an example to illustrate our results. p b Example 2.1. In Banach algebra A, if x A or x p 0 A) relative to the idempotent p A, then a c 0 d a p, a π 0, s 0 s, s π 1 p and w p aw aw) d. Using Theorem 2.1 or p b p 0 Theorem 2.2, we get that x A d and x d or x d ). c 0 References 1 N. Castro González, E. Dopazo, J. Robles, Formulas for the Drazin inverse of special block matrices, Appl. Math. Comput. 174 2006) 252 270. 2 N. Castro González, J.J. Koliha, New additive results for the g-drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 2004) 1085-1097. 3 D.S. Cvetkovic-Ilić, Y. Wei, Representations for the Drazin inverse of bounded operators on Banach space, Electron. J. Linear Algebra 18 2009) 613 627. 4 C. Deng, A note on the Drazin inverses with Banachiewicz-Schur forms, Appl. Math. Comput. 213 2009) 230 234. 5 C. Deng, D.S. Cvetkovic-Ilić, Y.Wei, Some results on the generalized Drazin inverse of operator matrices, Linear and Multilinear Algebra 58 2010) 503 521. 6 C. Deng, Y. Wei, Representations for the Drazin inverse of 2 2 blockoperator matrix with singular Schur complement, Linear Algebra and its Applications 435 2011) 2766 2783. 12
7 D.S. Djordjević, P.S. Stanimirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 2001) 617634. 8 D.S. Djordjević, Y. Wei, Additive results for the generalized Drazin inverse, J. Austral. Math. Soc. 73 2002) 115125. 9 R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2 2 block matrix, SIAM J. Matrix Anal. Appl. 27 2006) 757-771. 10 R.E. Hartwig, G. Wang and Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 2001) 207217. 11 J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 1996), 367 381. 12 M.Z. Kolundžija, D. Mosić, D.S. Djordjević, Generalized Drazin inverse of certain block matrices in Banach algebras, preprint). 13 X. Li, A representation for the Drazin inverse of block matrices with a singular generalized Schur complement, Appl. Math. Comput. 21718) 2011) 7531 7536. 14 X. Li, Y. Wei, A note on the representations for the Drazin inverse of 2 2 block matrix, Linear Algebra Appl. 423 2007) 332-338. 15 M.F. Martínez-Serrano, N. Castro-González, On the Drazin inverse of block matrices and generalized Schur complement, Appl. Math. Comput. 215 2009) 2733-2740. 16 J. Miao, Results of the Drazin inverse of block matrices, J. Shanghai Normal Univ. 18 1989) 25-31. 17 Y. Wei, Expressions for the Drazin inverse of 2 2 block matrix, Linear and Multilinear Algebra 45 1998) 131-146. Address: Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18000 Niš, Serbia E-mail D. Mosić: dijana@pmf.ni.ac.rs D. S. Djordjević: dragan@pmf.ni.ac.rs 13