Representation for the generalized Drazin inverse of block matrices in Banach algebras

Applied Mathematics and Computation 218 (2012) 12001–12007

Contents lists available at SciVerse ScienceDirect

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

Representation for the generalized Drazin inverse of block matrices in Banach algebras q Dijana Mosic´ ⇑, Dragan S. Djordjevic´ Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18000 Niš, Serbia

a r t i c l e

i n f o

Keywords: Generalized Drazin inverse Schur complement Block matrix

a b s t r a c t Several representations of the generalized Drazin inverse of a block matrix with a group invertible generalized Schur complement in Banach algebra are presented. Ó 2012 Elsevier Inc. All rights reserved.

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 [1,3–7,9,14,17]. 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 2 A, we use rðaÞ and qðaÞ, respectively, to denote the spectrum and the resolvent set of a. The sets of all nilpotent and quasinilpotent elements (rðaÞ ¼ f0g) of A will be denoted by Anil and Aqnil , respectively. The generalized Drazin inverse of a 2 A (or Koliha–Drazin inverse of a) is the element b 2 A which satisfies

bab ¼ b;

ab ¼ ba;

a  a2 b 2 Aqnil :

If the generalized Drazin inverse of a exists, it is unique and denoted by ad , and a is generalized Drazin invertible (see [11]). Recall that p ¼ 1  aad is the spectral idempotent of a corresponding to the set {0}, and it will be denoted by ap . The set of all generalized Drazin invertible elements of A is denoted by Ad . The Drazin inverse is a special case of the generalized Drazin inverse for which a  a2 b 2 Anil instead of a  a2 b 2 Aqnil . Obviously, if a is Drazin invertible, then it is generalized Drazin invertible. The group inverse is the Drazin inverse for which the condition a  a2 b 2 Anil is replaced with a ¼ 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 ¼ p2 2 A be an idempotent. Then we can represent element a 2 A as

a¼



a11

a12

a21

a22

 ;

where a11 ¼ pap, a12 ¼ pað1  pÞ, a21 ¼ ð1  pÞap, a22 ¼ ð1  pÞað1  pÞ. q

The authors are supported by the Ministry of Education and Science, Republic of Serbia, Grant No. 174007.

⇑ Corresponding author.

E-mail addresses: [email protected] (D. Mosic´), [email protected] (D.S. Djordjevic´). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.06.008

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 218 (2012) 12001–12007

12002

We use the following lemmas. Lemma 1.1. Let b 2 Ad and a 2 Aqnil . (i) [2, Corollary 3.4] If ab ¼ 0, then a þ b 2 Ad and ða þ bÞd ¼ P d nþ1 n (ii) If ba ¼ 0, then a þ b 2 Ad and ða þ bÞd ¼ 1 : n¼0 a ðb Þ

P1

d nþ1 n a : n¼0 ðb Þ

Specializing [2, Corollary 3.4] (with multiplication reversed) to bounded linear operators Castro-González and Koliha [2] recovered [8, Theorem 2.2] which is a spacial case of Lemma 1.1(ii). By the following lemma, Castro-González and Koliha [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 2 Ad and let ab ¼ 0. Then

ða þ bÞd ¼

1 1 X X d p n ðb Þnþ1 an ap þ b b ðad Þnþ1 : n¼0

n¼0

The following result is well-known for complex matrices (see [16]) and it is proved for elements of Banach algebra in [12].   a b d Lemma 1.3. [12, Lemma 2.2] Let x ¼ 2 A relative to the idempotent p 2 A, a 2 ðpApÞd and let w ¼ aad þ ad bca be such c d that aw 2 ðpApÞd . If cap ¼ 0, ap b ¼ 0 and the generalized Schur complement s ¼ d  cad b is equal to 0, then d

x ¼



p

0

cad

0

"

½ðawÞd 2 a 0 0

#"

0

p

ad b

0

0

# ð1Þ

:

Let

x¼



a b c

d

 2A

ð2Þ

relative to the idempotent p 2 A, a 2 ðpApÞd and let the generalized Schur complement s ¼ d  cad b 2 ðð1  pÞAð1  pÞÞ# . The generalized Schur complement s plays an important role in the representations for xd 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 ¼ AAD þ AD BCAD 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 ¼ d  cad 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 2 A, (ii) a 2 ðpApÞd , (iii) s ¼ d  cad b and s 2 ðð1  pÞAð1  pÞÞ# , p (iv) w ¼ aad þ ad bs cad . p

First, we give a formula for the generalized Drazin inverse of block matrix x in (2) in terms of w ¼ aad þ ad bs cad and the group invertible Schur complement s. Theorem 2.1. If aw 2 ðpApÞd ,

ap b ¼ 0; d

then x 2 A and

p

bs cap ¼ 0;

#

wbss ¼ 0;

#

ss# cad bss ¼ 0;

ð3Þ

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 218 (2012) 12001–12007

" xd ¼

1þ " þ

where

0

bs

#! 

#

#

#

p

bs



0

p þ s# cw½ðawÞd 2 a s#  s# cw½ðawÞd 2 bs

#

0 cad bs

0

1  X n¼1

12003

0

0

ðs# Þnþ1 can1 ap þ v n a

v n bsp

r;

#

0 ð1  pÞ  ds

!

ð4Þ

v n ¼ ðs# Þnþ1 cad ðawÞn1 ðawÞp , ðn ¼ 1; 2; . . .Þ, and "

r¼

#

p

½ðawÞd 2 a

½ðawÞd 2 bs

p

cad ½ðawÞd 2 a cad ½ðawÞd 2 bs

:

Proof. Using the equalities aad þ ap ¼ p, ss# þ sp ¼ 1  p and ssp ¼ 0, note that

" x¼

#

p

a2 ad

bs

caad

cad bs

" þ

p

#

aap

bss

cap

dss

# :¼ y þ z:

#

By ad ap ¼ 0 and (3), we obtain

" yz ¼

p

#

bs cap

awbss

p

cad bs cap

cwbss

# ¼ 0:

#

p

p

To check that y 2 Ad , set Ay  a2 ad , By  bs , C y  caad and Dy  cad bs . Since ða2 ad Þ# ¼ ad , Ay 2 ðpApÞ# and Sy  Dy 

C y A# y By

p

p

p

¼ 0. Also, Ay By ¼ ap bs ¼ 0, C y Ay ¼ 0 and W y ¼

Ay A# y

þ

# A# y By C y Ay

¼ w. Applying Lemma 1.3, observe that

d

y 2 A and

yd ¼



p

0

cad

0

"

½ðawÞd 2 a 0 0

#"

0

p

p

ad bs

0

0

# ¼ r:

In order to prove that z 2 Ad , we write

z¼



aap

0

cap

0

Recall that, for u ¼



 þ



m 0

0 0 0

 t , n

"



s

þ

#

#

0

bss

:¼ z1 þ z2 þ z3 :

#

0 cad bss

k 2 qpAp ðmÞ \ qð1pÞAð1pÞ ðnÞ ) k 2 qðuÞ; i.e.

rðuÞ # rpAp ðmÞ [ rð1pÞAð1pÞ ðnÞ: From aap 2 ðpApÞqnil and s 2 A# , we conclude that z1 2 Aqnil , z2 2 A# and z# 2 ¼ get z1 þ z2 2 Ad and ðz1 þ z2 Þd ¼ d

d

P1

# nþ1 n z1 : n¼0 ðz2 Þ



0 0

 0 # . Using Lemma 1.1(i), by z1 z2 ¼ 0, we s

Further, z23 ¼ 0, i.e. z3 2 Anil and ðz1 þ z2 Þz3 ¼ 0. By Lemma 1.1(ii), z 2 Ad and

d 2

z ¼ ðz1 þ z2 Þ þ z3 ½ðz1 þ z2 Þ  . Therefore, by Lemma 1.2, we deduce that x 2 Ad and

xd ¼

1 1 1 1 X X X X ðzd Þnþ1 yn yp þ zp zn ðyd Þnþ1 ¼ ð1 þ z3 ðz1 þ z2 Þd Þ½ðz1 þ z2 Þd nþ1 yn yp þ zp zn ðyd Þnþ1 :¼ X 1 þ X 2 ; n¼0

n¼0

n¼0

n¼0

where

X 1 ¼ ð1 þ z3 ðz1 þ z2 Þd Þx1 yp ;

x1 ¼

1 X ½ðz1 þ z2 Þd nþ1 yn ; n¼0

X 2 ¼ zp yd þ

1 X

zp zn ðyd Þnþ1 :

n¼1

Since z1 y ¼ 0, then ðz1 þ z2 Þd y ¼



0 0

 0 # # y ¼ z2 y and s

1 1 1 X X X d kþ1 k n x1 ¼ ðz1 þ z2 Þ þ ½ðz1 þ z2 Þd n z# z# ðz# z1 2 y ¼ ðz1 þ z2 Þ þ 2 þ 2Þ

!n

d

n¼1

¼ ðz1 þ z2 Þd þ

1 X n¼1

n¼1

n # n1 ðz# 2 Þ þ ðz2 Þ

k¼1

n z# 2y

! 1 1 1 X X X kþ1 k nþ1 n nþ1 n # n ðz# Þ z ðz# z1 þ ðz# y : 1 z2 y ¼ 2 2Þ 2Þ k¼1

n¼0

n¼1

ð5Þ

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 218 (2012) 12001–12007

12004

Now, we have

! ! 1 1 1 X X 0 b X # kþ1 k # nþ1 n # nþ1 n yp ðz Þ z ðz Þ z þ ðz Þ y 1 1 2 2 0 cad b k¼0 2 n¼0 n¼1 #! !  " 1 1 1 X X X 0 b # # kþ1 k # # n n # n n z2 þ 1þ ðz2 Þ z1 z2 ðz2 Þ z1 þ ðz2 Þ y yp 0 cad b n¼0 n¼1 k¼1 " #! ! # 1 1 X X 0 bs # p # nþ1 n p # nþ1 n p 1þ ðz2 Þ z1 y þ ðz2 Þ y y z2 y þ # 0 cad bs n¼1 n¼1 " #! ! # 1 1 X X 0 bs # p # nþ1 n # nþ1 n p z2 y þ ðz2 Þ z1 þ ðz2 Þ y y : 1þ # 0 cad bs n¼1 n¼1 

X1 ¼ 1 þ ¼ ¼ ¼

ð6Þ

Observe that aad ðawÞ ¼ aw ¼ ðawÞaad ,

" p

d

y ¼ 1  yy ¼

#

p

p  ðawÞd a

ðawÞd bs

p

cw½ðawÞd 2 a ð1  pÞ  cw½ðawÞd 2 bs

and

" n

p

y y ¼

#

p

aad ðawÞn1 ðawÞp a

ðawÞn1 ðawÞp bs

cad ðawÞn1 ðawÞp a

cad ðawÞn1 ðawÞp bs

p

ðn ¼ 1; 2; . . .Þ:

Hence, these equalities and (6) give

"

#! 

 n p 1  X a a 0 0 þ p # # nþ1 n1 p s# cw½ðawÞd 2 a s#  s# cw½ðawÞd 2 bs 0 ðs ca Þ a 0 cad bs n¼1 " # ! !   # 1 X 0 0 0 bs þ ¼ 1þ n1 p n1 p p # # nþ1 d # nþ1 d d ðs Þ ca ðawÞ ðawÞ a ðs Þ ca ðawÞ ðawÞ bs 0 ca bs n¼1   X ! 1  0 0 0 0  : p þ p ðs# Þnþ1 can1 ap þ v n a v n bs s# cw½ðawÞd 2 a s#  s# cw½ðawÞd 2 bs n¼1

X1 ¼

1þ

#

0

bs

0



0

0



0

ð7Þ

From # # zzd y ¼ ½ðz1 þ z2 Þðz1 þ z2 Þd þ z3 ðz1 þ z2 Þd y ¼ ðz1 þ z2 Þz# 2 y þ z3 z2 y ¼ ðz2 þ z3 Þz2 y;

   # bs p d we get zzd yd ¼ ðz2 þ z3 Þz# and zp yd ¼ p # r. Thus, by z 2y 0 ð1  pÞ  ds 0   # bs aw½ðawÞd 2 ¼ 0, notice that zzp yd ¼ z p # r ¼ z1 r ¼ 0 and 0 ð1  pÞ  ds

X 2 ¼ zp yd þ

#

bs # ð1  pÞ  ds



¼ z1 and ap ðawÞd ¼ ap aad

1 X zn zp ðyd Þnþ1 ¼ zp r:

ð8Þ

n¼1

So, (5), (7) and (8) imply (4).

h

p # # Our conditions ap b ¼ 0, bs cap ¼ 0, wbss ¼ 0, ss# cad bss ¼ 0 in Theorem 2.1 can be formulated geometrically as

bA  aA;



sp cap A  b ;

#

bss A  w ;

#

cad bss A  s ;

where e ¼ ff 2 A : ef ¼ 0g. If we assume that a is a group invertible and w ¼ 1 in Theorem 2.1, we obtain the following result as a consequence. # p # Corollary 2.1. Let a 2 ðpApÞ# and let w ¼ aa# þ a# bca ¼ 1. If ap b ¼ 0, bca ¼ 0 and bss ¼ 0; then x 2 Ad and

"

a#

ða# Þ2 b

s# ca# þ ðs# Þ2 cap þ sp cða# Þ2

s#  s# cða# Þ2 b þ sp cða# Þ3 b

d

x ¼

# :

In the following theorem, the other formula for the generalized Drazin inverse of block matrix is presented. Theorem 2.2. If aw 2 ðpApÞd ,

cap ¼ 0;

p

ap bs c ¼ 0;

ss# cw ¼ 0;

#

ss# cad bss ¼ 0;

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 218 (2012) 12001–12007

12005

then x 2 Ad and

" xd ¼

#

ðawÞd bs

#

0

0 s#  sp cad ðawÞd bs   p 0

þ

#

þt

s# c

ð1  pÞ  s# d

" #!  1 X 0 0 an1 ap bðs# Þnþ1 þ aad ðawÞn1 ðawÞp bðs# Þnþ1 1 þ # p cad ðawÞn1 ðawÞp bðs# Þnþ1 s c 0 s n¼1

0



s# cad b ð9Þ

;

where

"

½ðawÞd 2 a

t¼

#

½ðawÞd 2 b

sp cad ½ðawÞd 2 a sp cad ½ðawÞd 2 b

:

Proof. If we write

" x¼

a2 ad

aad b

sp c

sp cad b

#

 þ

aap

ap b

ss# c

ss# d



:¼ y þ z;

then zy ¼ 0. Using Lemma 1.3, we conclude that y 2 Ad and yd ¼ t. To show that z 2 Ad , let

z¼



aap

ap b

0

0



 þ

0 0 0



 þ

s



0

0

ss# c

ss# cad b

:¼ z1 þ z2 þ z3 :

Then z1 2 Aqnil , z2 2 A# and z2 z1 ¼ 0. By Lemma 1.1(ii), z1 þ z2 2 Ad and ðz1 þ z2 Þd ¼ d

d

d

P1

n # nþ1 . n¼0 z1 ðz2 Þ

From z23 ¼ 0,

d 2

z3 ðz1 þ z2 Þ ¼ 0 and Lemma 1.1(i), z 2 A and z ¼ ðz1 þ z2 Þ þ ½ðz1 þ z2 Þ  z3 . Applying Lemma 1.2, observe that x 2 Ad and 1 1 X X ðyd Þnþ1 zn zp þ yp yn ðzd Þnþ1 :¼ X 1 þ X 2 :

xd ¼

n¼0

ð10Þ

n¼0

d p The equalities yz1 ¼ 0 and yðz1 þ z2 Þd ¼ yz# 2 imply y z ¼ t

X 1 ¼ yd zp þ

1 X



p s# c

0 ð1  pÞ  s# d



and yd zp z ¼ 0.

ðyd Þnþ1 zp zn ¼ tzp :

ð11Þ

n¼1

Next, we obtain

X2 ¼

1 1 X X yp yn ðzd Þnþ1 ¼ yp yn ½ðz1 þ z2 Þd nþ1 ð1 þ ðz1 þ z2 Þd z3 Þ n¼0

n¼0

! 1 X d d n p p n # y ðz1 þ z2 Þ þ y y z2 ½ðz1 þ z2 Þ  ð1 þ ðz1 þ z2 Þd z3 Þ

¼

n¼1

"

1 1 1 X X X nþ1 n kþn þ y p y n z# ðz# zk1 ðz# ¼ yp zn1 ðz# 2Þ 2 2Þ þ 2Þ

"

n¼0

n¼1

n¼0

n¼1

!# 1þ

z# 2 z3

1 X nþ1 þ zn1 ðz# z3 2Þ

!

n¼1

k¼1

# " # 1 1 1 1 X X X X nþ1 # p n # nþ1 p # n # nþ1 p n # nþ1 ¼ yp zn1 ðz# Þ þ y y ðz Þ z Þ ¼ y z þ z ðz Þ þ y y ðz Þ ð1 þ z ð1 þ z# 1 2 2 2 2 3 2 2 2 z3 Þ: " Since yp ¼

p  ðawÞd a sp cad ðawÞd a

" yn yp ¼

n¼1

ðawÞd b ð1  pÞ  sp cad ðawÞd b

aad ðawÞn1 ðawÞp a

n¼1

# and

aad ðawÞn1 ðawÞp b

sp cad ðawÞn1 ðawÞp a sp cad ðawÞn1 ðawÞp b

# ðn ¼ 1; 2; . . .Þ;

we get

" X2 ¼

0

#

ðawÞd bs

#

0 s#  sp cad ðawÞd bs

#

þ

" #!  1 X 0 0 an1 ap bðs# Þnþ1 þ aad ðawÞn1 ðawÞp bðs# Þnþ1 1 þ # p cad ðawÞn1 ðawÞp bðs# Þnþ1 s c 0 s n¼1

0



s# cad b ð12Þ

Thus, from (10)–(12), we obtain (9). h

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 218 (2012) 12001–12007

12006

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 2 ðpApÞ# and w ¼ 1, the next corollary follows. # Corollary 2.2. Let a 2 ðpApÞ# and let w ¼ aa# þ a# bca ¼ 1. If cap ¼ 0, ap bc ¼ 0 and ss# c ¼ 0, then x 2 Ad and

" x ¼

p

a#

a# bs þ ap bðs# Þ2 þ ða# Þ2 bs

sp cða# Þ2

s#  cða# Þ2 bs þ sp cða# Þ3 bs

d

#

# :

p

#

The following result is a consequence of Theorems 2.1 and 2.2. Corollary 2.3. If aw 2 ðpApÞd , s ¼ 0 and if one of the following conditions holds: (i) (ii) (iii)

ap b ¼ 0 and cap ¼ 0, p ap b ¼ 0 and bca ¼ 0, cap ¼ 0 and ap bc ¼ 0,

then x 2 Ad and

"

d

x ¼

½ðawÞd 2 a

#

½ðawÞd 2 b

cad ½ðawÞd 2 a cad ½ðawÞd 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 ¼ 0 in Theorem 2.1, we obtain the next result.   a 0 Theorem 2.3. Let x ¼ 2 A relative to the idempotent p 2 A, a 2 ðpApÞd and s 2 ðð1  pÞAð1  pÞÞ# . Then c s (i) x 2 Ad and

"

ad

0

sp cðad Þ2  s# cad

s#

d

x ¼

#

1  X

þ

n¼1

0

0

ðs# Þnþ1 can1 ap

0

 ;

ð13Þ

(ii) x 2 A# if and only if a 2 ðpApÞ# and sp cap ¼ 0. Furthermore, if a 2 ðpApÞ# and sp cap ¼ 0, then

x# ¼



a# sp cða# Þ2

#

0 # 2

#

 s ca þ ðs Þ

cap

#

s

 :

Proof. (i) If b ¼ 0 in Theorem 2.1, then s ¼ d, w ¼ aad , aw ¼ a2 ad 2 ðpApÞ# , ðawÞ# ¼ ad and ad ðawÞp ¼ ad ap ¼ 0 implying (13). (ii) By the part (i), note that

2 6 x2 xd ¼ 4

a2 ad caad þ

0

1 X s2 ðs# Þnþ1 can1 ap

3

7 : s5

n¼1

Consequently, x2 xd ¼ x () a2 ad ¼ a and caad þ ss# cap ¼ cap . h

P1

# n n1 p a n¼1 sðs Þ ca

¼ c. Hence, x 2 A# is equivalent to a 2 ðpApÞ# and

In the same manner as in the proof of Theorem 2.3, if c ¼ 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 ¼ 2 A relative to the idempotent p 2 A, a 2 ðpApÞd and s 2 ðð1  pÞAð1  pÞÞ# . Then 0 s (i) x 2 Ad and

" xd ¼

p

ad

ðad Þ2 bs  ad bs

0

s#

#

# þ

" # 1 X 0 an1 ap bðs# Þnþ1 n¼1

0

0

;

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 218 (2012) 12001–12007

12007

p p (ii) x 2 A# if and only if a 2 ðpApÞ# and ap bs ¼ 0. Furthermore, if a 2 ðpApÞ# and ap bs ¼ 0, then

" x# ¼

p

a#

ða# Þ2 bs  a# bs þ ap bðs# Þ2

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 0 p b Example 2.1. In Banach algebra A, if x ¼ 2 A (or x ¼ 2 A) relative to the idempotent p 2 A, then ad ¼ a ¼ p, c 0 0 0 ap ¼ 0, s ¼ 0 ¼ s# , sp ¼ 1  p and w ¼ p ¼ aw ¼ ðawÞd . Using Theorem 2.1 or Theorem 2.2, we get that x 2 Ad and     p 0 p b or xd ¼ . xd ¼ c 0 0 0 Acknowledgement We are grateful to the anonymous referees. Their helpful suggestions led to the improved presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

N. Castro-González, E. Dopazo, J. Robles, Formulas for the Drazin inverse of special block matrices, Appl. Math. Comput. 174 (2006) 252–270. N. Castro-González, J.J. Koliha, New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 1085–1097. D.S. Cvetkovic´-Ilic´, Y. Wei, Representations for the Drazin inverse of bounded operators on Banach space, Electron. J. Linear Algebra 18 (2009) 613–627. C. Deng, A note on the Drazin inverses with Banachiewicz-Schur forms, Appl. Math. Comput. 213 (2009) 230–234. C. Deng, D.S. Cvetkovic´-Ilic´, Y. Wei, Some results on the generalized Drazin inverse of operator matrices, Linear Multilinear Algebra 58 (2010) 503–521. C. Deng, Y. Wei, Representations for the Drazin inverse of 2  2 block-operator matrix with singular Schur complement, Linear Algebra Appl. 435 (2011) 2766–2783. D.S. Djordjevic´, P.S. Stanimirovic´, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (2001) 617–634. D.S. Djordjevic´, Y. Wei, Additive results for the generalized Drazin inverse, J. Austral. Math. Soc. 73 (2002) 115–125. 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. R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207–217. J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996) 367–381. M.Z. Kolundzˇija, D. Mosic´, D.S. Djordjevic´, Generalized Drazin inverse of certain block matrices in Banach algebras, preprint. X. Li, A representation for the Drazin inverse of block matrices with a singular generalized Schur complement, Appl. Math. Comput. 217 (18) (2011) 7531–7536. 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. 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. J. Miao, Results of the Drazin inverse of block matrices, J. Shanghai Normal Univ. 18 (1989) 25–31. Y. Wei, Expressions for the Drazin inverse of 2  2 block matrix, Linear Multilinear Algebra 45 (1998) 131–146.