New additive results on Drazin inverse and its applications

Applied Mathematics and Computation 218 (2011) 3019–3024

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

New additive results on Drazin inverse and its applications q Dragana S. Cvetkovic´-Ilic´ University of Niš, Department of Mathematics, Faculty of Science, P.O. Box 224, Višegradska 33, 18000 Niš, Serbia

a r t i c l e

i n f o

a b s t r a c t In this paper, the Drazin inverse of certain sums and block triangular matrices are presented. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Banach algebra Drazin inverse Additive properties Operator block-matrix Bounded operator

1. Results Let A be a complex Banach algebra with the unit 1. Let us recall that the Drazin inverse of a 2 A [9] is the element x 2 A (denoted by ad) which satisfies

xax ¼ x;

ax ¼ xa;

akþ1 x ¼ ak ;

ð1:1Þ

for some nonnegative integer k. The least such k is the index of a, denoted by ind(a). When ind (a) = 1, then the Drazin inverse ad is called the group inverse and is denoted by ag or a#. The set Ad consists of all a 2 A such that ad exists. The theory of Drazin inverses has seen a substantial growth over the past few decades. It is a subject which is of great theoretical interest and finds applications in a great many of various areas, including statistics, numerical analysis, differential equations, Markov chains, population models, cryptography, and control theory. One particular topic concerning Drazin inverses that is of considerable interest is finding explicit representations for the Drazin inverse of a sum of two elements and explicit representations for the Drazin inverse of a 2  2 block matrix. There have been many papers on this subject but the motivation for this paper was the paper of Patricio and Hartwig [14] in which the authors presented a formula for the Drazin inverse of the sum of two elements of a Banach algebra under some conditions. In this paper we generalize their results and obtain some new additive results on the Drazin inverse of a sum of two elements in a Banach algebra which are in general not Drazin invertible. As an application of our results we get generalizations of almost all the results from [1,6]. If p ¼ p2 2 A is an idempotent then we can represent elements a 2 A as an operator matrix

 a p

a11

a12

a21

a22

 ;

ð1:2Þ

where a11 = pap, a12 = pa(1  p), a21 = (1  p)ap, a22 = (1  p)a(1  p). First we state the following result which was proved in [13] for matrices, extended in [10] for bounded linear operators and in [5] for arbitrary elements of a Banach algebra.

q

Supported by Grant No. 174007 of the Ministry of Science, Technology and Development, Republic of Serbia. E-mail address: [email protected]

0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.083

D.S. Cvetkovic´-Ilic´ / Applied Mathematics and Computation 218 (2011) 3019–3024

3020

Theorem 1.1. Let x; y 2 A and let

x p



 a c ; 0 b

y 1p



b 0 c



a

relative to the idempotent p 2 A. (1) If a 2 ðpApÞd and b 2 ðð1  pÞAð1  pÞÞd , then x and y are Drazin invertible and d

x p



 d ; b

ad

u

0

y 1p

!

d

0

u

ad

b

d

ð1:3Þ

;

P P n p d nþ2 d d nþ2 p n where u ¼ 1 cb b þ 1  ad cb and ap = 1  aad. n¼0 ða Þ n¼0 a a cðb Þ d d (2) If x 2 A and a 2 ðpApÞ , then b 2 ðð1  pÞAð1  pÞÞd and xd, yd are given by (1.3). We introduce the following notations which will be used throughout the paper. For a; b 2 A and i 2 N n f1g defined by

Pi ¼

( i1 Y

k

ðajs b s Þ :

s¼1

Si ¼

X

i1 X ðjs þ ks Þ ¼ i  1;

) j1 ; . . . ; ji1 ; k1 ; . . . ; ki1 2 f0; 1; . . . ; i  1g ;

s¼1

ð1:4Þ

t ¼ ða þ bÞi1 :

t2P i

In the case when i = 1, let P1 = {1} and S1 = 1. Theorem 1.2. Let a; b 2 A and i 2 N be such that aSi ; Si b 2 Ad and aSib = 0. Then a þ b 2 Ad and (a + b)d is represented by the following (1) if i = 1, then d

d

ða þ bÞd ¼ ad þ bX 1 ad a þ bb X 1 a þ b ;

ð1:5Þ

(2) if i = 2, then

ða þ bÞd ¼ ðaS2 Þd a þ bX 2 a þ bðS2 bÞd ;

ð1:6Þ

(3) if i > 2, then

ða þ bÞd ¼ aSi2 ðaSi Þd a þ aSi2 bX i a þ bSi2 ðaSi Þd a þ bSi2 bX i a þ aSi2 bðSi bÞd þ bSi2 bðSi bÞd ;

ð1:7Þ

where

Xi ¼

indðaS Xi Þ

ððSi bÞd Þnþ2 Si ðaSi Þn ðaSi Þp þ

n¼0

Mi ¼

a 1



ð1:8Þ

  a , using Cline’s formula [7] bÞ 1

ða þ bÞd ¼ ð 1 b ÞðM d Þ2 

ðSi bÞp ðSi bÞn Si ððaSi Þd Þnþ2  ðSi bÞd Si ðaSi Þd :

n¼0

Proof. Since a þ b ¼ ð 1

where M ¼

indðS Xi bÞ

  a ; 1

 ab . Using mathematical induction, we prove that b

aSi

aSi b

Si

Si b

 ;

where Si is defined by (1.4) and i 2 N. Since aSi ; Si b 2 Ad and aSib = 0 by Theorem 1.1, we get that i d

d i

ðM Þ ¼ ðM Þ ¼

ðaSi Þd

0

Xi

ðSi bÞd

!

;

where Xi is defined by (1.8). Now, for i = 1, we get that

ða þ bÞd ¼ ð 1 b Þ

ad

0

X1

b

d

!2   a 1

d

d

d

d

¼ ðad Þ2 a þ bX 1 ad a þ bb X 1 a þ bðb Þ2 ¼ ad þ bX 1 ad a þ bb X 1 a þ b ;

D.S. Cvetkovic´-Ilic´ / Applied Mathematics and Computation 218 (2011) 3019–3024

3021

while in the case when i = 2, we get

ða þ bÞd ¼ ð 1 b Þ

!  a

ðaS2 Þd

0

X2

ðS2 bÞd

1

¼ ðaS2 Þd a þ bX 2 a þ bðS2 bÞd :

For all other cases (i > 2), we have that

ða þ bÞd ¼ ð 1 b ÞM i2 ðMd Þi

  a ; 1

so

ða þ bÞd ¼ ð 1 b Þ



aSi2

aSi2 b

Si2

Si2 b



!  a

ðaSi Þd

0

Xi

ðSi bÞd

d

1

d

¼ aSi2 ðaSi Þ a þ aSi2 bX i a þ bSi2 ðaSi Þ a þ bSi2 bX i a þ aSi2 bðSi bÞd þ bSi2 bðSi bÞd :



The part (1) of the Theorem 1.2 (i = 1) derived a representation of (a + b)d under the condition ab = 0 and it is actually the main results from [12] where a representation of (a + b)d is given by the following equivalent form to those given by (1.5): indðbÞ X

d

ða þ bÞd ¼ ð1  bb Þ

!

r

b ðad Þr ad þ b

indðaÞ X

d

r¼0

!

d

ðb Þr ar ð1  aad Þ:

r¼0

In the case i = 2, Theorem 1.2 (2) provided a representation of (a + b)d under the condition a2b + ab2 = 0. Remark that Theorem 2.3 from the paper of Castro González et al. [6] is a special case of that result: Theorem 1.3 [6]. Let F; G 2 BðX Þ be Drazin invertible such that G2F = GF2 = 0 and let GF be Drazin invertible. Then F + G is Drazin invertible and

ðF þ GÞd ¼ UGp þ F p V þ XðI þ YGÞGp þ F p ðI þ FXÞY þ FUV þ UVG þ

2rþt2 X

ðF d Þkþ1 Ckþ2 G þ

k¼0

2rþs2 X

FBkþ2 ðGd Þkþ1 ;

k¼0

where X, Y, U, V, Bk+2, Ck+2 are defined as in Lemma 2.2 of [6]. For i = 3 we get a new result: 2

3

2

Corollary 1.1. Let a; b 2 A be such that a3 þ ab þ a2 b þ aba; a2 b þ b þ ab þ bab 2 Ad and a3b + ab3 + a2b2 + (ab)2 = 0. Then a þ b 2 Ad and 2

2

3

2

ða þ bÞd ¼ aða3 þ ab þ a2 b þ abaÞd a þ abX 3 a þ bða3 þ ab þ a2 b þ abaÞd a þ bbX 3 a þ abða2 b þ b þ ab þ babÞd 2

3

2

þ b ða2 b þ b þ ab þ babÞd ; where

X3 ¼

indðaS X3 Þ n¼0

ððS3 bÞd Þnþ2 S3 ðaS3 Þn ðaS3 Þp þ

indðS X3 bÞ

ðS3 bÞp ðS3 bÞn S3 ððaS3 Þd Þnþ2  ðS3 bÞd S3 ðaS3 Þd

ð1:9Þ

n¼0

and S3 = (a + b)2. The following result is a generalization of Corollary 3.1 from [14]: Theorem 1.4. Suppose that aða þ bÞi2 a; aða þ bÞi2 b; bða þ bÞi2 b 2 Ad and aPib = {0}, where i 2 N n f1g. Then a þ b 2 Ad and the Drazin inverse is given by (1.6) if i = 2 or by (1.7) if i > 2 and it can be expressed in terms of a,b and a(a + b)i2a, a(a + b)i2b, b(a + b)i2b and their Drazin inverses. Proof. Denote by Ea the sum of all the elements of Pi ending with a and by Eb the sum of all the elements of Pi ending with b. P It is evident that Si = Ea + Eb and (aEa)(aEb) = 0, so ðaSi Þn ¼ nk¼0 ðaEb Þk ðaEa Þnk . Furthermore, Ea = (a + b)i2a and Eb = (a + b)i2b. Similarly, if we denote by Ba the sum of all the elements of Pi beginning with a and by Bb the sum of all the elements of Pi P beginning with b. We have that Si = Ba + Bb and (Bab)(Bbb) = 0, so ðSi bÞn ¼ nk¼0 ðBb bÞk ðBa bÞnk . Similarly, Ba = a(a + b)i2 and i2 Bb = b(a + b) . Now, by Theorem 2.3 [11], we get that (aSi)d and (Sib)d are Drazin invertible and that their Drazin inverses can be represented in terms of aEa, aEb, Bab and Bbb. Hence, the result follows by Theorem 1.2. h In the special case when i = 2, we get the next result. Remark that in Corollary 3.1 of [14] the condition that a2 + ab and ab + b2 be Drazin invertible is unnecessary. Corollary 1.2 [14]. Suppose that a; b; ab 2 Ad and a2b = ab2 = 0. Then a + b has a Drazin inverse given by (1.6) and it can be expressed in terms of a, b, ad, bd and (ab)d.

D.S. Cvetkovic´-Ilic´ / Applied Mathematics and Computation 218 (2011) 3019–3024

3022

2. Applications As an application of Theorem 1.2 we obtain representations for the Drazin inverse of an operator matrix which extend some of the already known results. The problem of finding explicit representations for the Drazin inverse of a 2  2 block   A B matrix M ¼ in terms of the blocks of the partition was posed by Campbell and Meyer [4]. Such representations C D are, in particular, very useful to find a general expression for the solutions of a second order system of the differential equations (see [2,3,12]). Until now, there has been no explicit formula for the Drazin inverse of M in terms of Ad and Dd with arbitrary A, B, C and D. Let Y; Z be two Banach spaces and X ¼ Y  Z. We will suppose that

 M¼



A

B

C

D

ð2:10Þ

is a bounded linear operator matrix on X . The following theorem is a generalization of Theorems 4.2 and 4.4 of [6] in which the additional conditions: BC is nilpotent and DC = 0, respectively, are required. Theorem 2.1. Let M given by (2.10) be such that A 2 BðYÞ, D 2 BðZÞ and BC are Drazin invertible. If BCA = 0 and BD = 0, then M is Drazin invertible and

Md ¼

!

AðA2 þ BCÞd

ðBCÞd B þ AX 1 B

CðA2 þ BCÞd þ DU

CX 1 B þ DX 3 B þ Dd

ð2:11Þ

;

where 1 X

X1 ¼

! ððA2 þ BCÞd Þnþ2 AðBCÞn ðBCÞp þ ðA2 þ BCÞp ðA2 þ BCÞn AððBCÞd Þnþ2

 ðA2 þ BCÞd AðBCÞd

n¼0 1 X nþ2 X ð ðDd Þ2k UððA2 þ BCÞd Þnþ1k AðBCÞn ðBCÞp Þ þ ðDd Þ2n CðBCÞn ðBCÞp

 X3 ¼ 

1 X

"

!

n¼0 k¼0

n1   X  ðCA þ DCÞðA2 þ BCÞd  D2 U ðA2 þ BCÞn þ Dp D2k ðCA þ DCÞðA2 þ BCÞnk1

n¼0

!# AððBCÞd Þnþ2

k¼0

 þDp D2n CððBCÞd Þnþ2  UAðBCÞd  ðDd Þ2 CðBCÞd and

U¼

1 1 X X ððD2 Þd Þnþ2 ðCA þ DCÞðA2 þ BCÞn ðA2 þ BCÞp þ ðD2 Þp ðD2 Þn ðCA þ DCÞðð0A2 þ BCÞd Þnþ2 n¼0

n¼0

 ðD2 Þd ðCA þ DCÞðA2 þ BCÞd : Proof. Let M = P + Q, where P ¼

P1 ¼ PM ¼



BC

0

0

0

 ;



0 0

B 0

ð2:12Þ 

and Q ¼

Q 1 ¼ MQ ¼



A C

 0 . We have that D

A2 þ BC

0

CA þ DC

D2

! and PMQ ¼ 0:

Since (BC)A2 = 0, by Theorem 2.3 [11] we conclude that A2 + BC is Drazin invertible, so Q1 is Drazin invertible. The Drazin invertibility of P1 is evident. Now, by the special case of Theorem 1.2 when i = 2, we get that

Md ¼ Pd1 P þ QXP þ QQ d1 ; where

X¼

1 1 X X ðQ d1 Þnþ2 MPn1 P p1 þ Q p1 Q n1 MðPd1 Þnþ2  Q d1 MPd1 : n¼0

n¼0

By Theorem 1.1, we have that

Q d1

¼

ðA2 þ BCÞd

0

U

ðD2 Þd

where U is given by (2.12).

! ;

D.S. Cvetkovic´-Ilic´ / Applied Mathematics and Computation 218 (2011) 3019–3024

3023

So, by mathematical induction we can prove that

0

B Q n1 ¼ @ n1 P

ðA2 þ BCÞn

1

0

D2k ðCA þ DCÞðA2 þ BCÞnk1

D2n

ððA2 þ BCÞd Þn

0

C A

k¼0

and

0 B ðQ d1 Þn ¼ @ n1 P

ðDd Þ2k UððA2 þ BCÞd Þnk1

1

ðDd Þ2n

C A:

k¼0

Now, by computation we get that (2.11) holds. h Let us remark that by Theorem 2.3 from [11] the Drazin inverse of (A2 + BC) can be represented by

ðA2 þ BCÞd ¼ Ap

½indðAÞ=2þ1 X 

 indðBCÞ  X  d 2ðrþ1Þ A2r ððBCÞd Þrþ1 þ ðA Þ ðBCÞr ðBCÞp :

r¼0

r¼0

   0 B A 0 and Q ¼ under the 0 D C 0 conditions that BCA = 0 and DC = 0 we can show that M is Drazin invertible and obtain a representation of its Drazin inverse. This result is a generalization of Theorem 4.5 of [6]. The next result presents a generalization of Theorems 1–3 from [1] in which the additional conditions: BD = 0, BC is nilpotent and D is nilpotent, respectively are required. In a similar way as it was done in the previous theorem, now using the splitting P ¼



Theorem 2.2. Let M given by (2.10) be such that A 2 BðYÞ; D 2 BðZÞ and BC are Drazin invertible. If ABC = 0 and DC = 0, then M is Drazin invertible and

Md ¼

ðA2 þ BCÞd A

ðA2 þ BCÞd B þ UD

!

CðBCÞd þ CX 1 A CX 1 B þ CX 2 D þ Dd

ð2:13Þ

;

where

U¼

1 1 X X ðA2 þ BCÞnþ2 ðAB þ BDÞD2n Dp þ ðA2 þ BCÞp ðA2 þ BCÞn ðAB þ BDÞD2nþ4  ðA2 þ BCÞd ðAB þ BDÞðD2 Þd ; n¼0

X1 ¼

1  X

n¼0

 ððBCÞd Þnþ2 AðA2 þ BCÞn ðA2 þ BCÞp þ ðBCÞp ðBCÞn AððA2 þ BCÞd Þnþ2  ðBCÞd AðA2 þ BCÞd ;

n¼0

X2 ¼

1  X

  ððBCÞd Þnþ2 AðA2 þ BCÞn ðA2 þ BCÞU  ðAB þ BDÞðDd Þ2

n¼0 d nþ2

þððBCÞ Þ

! ! n1 nþ1   X X 2 i 2ðnþ1iÞ 2n Dp þ ðBCÞp ðBCÞn A A ½ðA þ BCÞ ðAB þ BDÞD  þ BD ððA2 þ BCÞd Þi UðDd Þ2ðniÞ i¼0

i¼0

þðBCÞp ðBCÞn BðDd Þ2nþ4 þ ðBCÞd AU þ ðBCÞd BðDd Þ2 : Proof. Let M = P + Q, where P ¼

P1 ¼ PM ¼

 0

   B 0 0 and Q ¼ . We have that D! C 0 

A2 þ BC

AB þ BD

0

D2

;

Q 1 ¼ MQ ¼

BC

0

0

0

and PMQ ¼ 0:

Similarly as in the proof of Theorem 2.1 we get that P1 and Q1 are Drazin invertible operators and, by Theorem 1.2, that the operator M is also Drazin invertible. Furthermore,

Md ¼ Pd1 P þ QXP þ QQ d1 ; where

X¼

1 1 X X  nþ2 ðQ d1 Þnþ2 MPn1 Pp1 þ Q p1 Q n1 M Pd1  Q d1 MP d1 : n¼0

n¼0

Now, by computation we get that (2.13) holds. h Let us remark that Theorem 2.2 also generalized the result from Corollary 2.3 [8].

3024

D.S. Cvetkovic´-Ilic´ / Applied Mathematics and Computation 218 (2011) 3019–3024

Acknowledgments The author would like to thank the anonymous referees for their useful suggestions, which helped to improve the original version of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

A.S. Cvetkovic´, G.V. Milovanovic´, On Drazin inverse of operator matrices, J. Math. Anal. Appl. 375 (2011) 331–335. S.L. Campbell, Singular Systems of Differential Equations, Pitman, London, 1980. S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear Multilinear Algebra 14 (1983) 195–198. S.L. Campbell, C.D. Meyer, Generalized Inverse of Linear Transformations, Pitman, London, 1979. Dover, New York, 1991. N. Castro González, J.J. Koliha, New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 1085–1097. N. Castro González, E. Dopazo, M.F. Martínez-Serrano, On the Drazin inverse of the sum of two operators and its application to operator matrices, J. Math. Anal. Appl. 350 (2009) 207–215. R.E. Cline, An Application of Representation of a Matrix, MRC Technical Report, 1965, p. 592. E. Dopazo, M.F. Martínez-Serrano, Further results on the Drazin inverse of a 2  2 block matrix, Linear Algebra Appl. 432 (2010) 1896–1904. M.P. Drazin, Pseudoinverse in associative rings and semigroups, Am. Math. Monthly 65 (1958) 506–514. D.S. Djordjevic´, P.S. Stanimirovic´, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (126) (2001) 617–634. D.S. Djordjevic´, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 72 (2002) 1–11. R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverses, Linear Algebra Appl. 322 (1–3) (2001) 207–217. C.D. Meyer Jr., N.J. Rose, The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. Math. 33 (1) (1977) 1–7. P. Patricio, R.E. Hartwig, Some additive results on Drazin inverses, Appl. Math. Comput. 215 (2009) 530–538.