Filomat 30:8 (2016), 2289–2294
DOI 10.2298/FIL1608289L
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
New Additive Results for the Generalized Drazin Inverse
in a Banach Algebra
Xiaoji Liua , Xiaolan Qinb , Julio Benı́tezc
author. Tel. +86-0771-3265356. Faculty of Science, Guangxi. University for Nationalities, Nanning 530006, P.R. China.
b Faculty of Science, Guangxi University for Nationalities, Nanning 530006, P.R. China.
c Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Valencia, camino de Vera s/n, 46022, Spain.
a Corresponding
Abstract. In this paper, we investigate additive properties of the generalized Drazin inverse in a Banach
algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new
conditions on a, b ∈ A.
1. Introduction
Let A be a complex Banach algebra with the unit 1. By A−1 and Aqnil , we denote the sets of all invertible
and quasinilpotent elements in A, respectively. In a Banach algebra A, an element a ∈ A is quasinilpotent
when limn→∞ kan k1/n = 0. Let us recall that the spectral radius of b ∈ A is given by r(b) = limn→∞ kbn k1/n (see
e.g. [1, Ch. 1]) and it is satisfied that r(b) = max{|λ| : λ ∈ σ(b)}, where σ(b) is the spectrum of b, i.e., the set
composed of complex numbers λ such that b − λ1 is not invertible.
Let us recall that a generalized Drazin inverse of a ∈ A (introduced by Koliha in [8]) is an element x ∈ A
which satisfies
xax = x,
ax = xa,
a − a2 x ∈ Aqnil .
(1)
It can be proved that for a ∈ A the set of x ∈ A satisfying (1) is empty or a singleton ([8]). If this set is a
singleton, then we say that a is generalized Drazin invertible and x is denoted by ad . The set Ad consits of
all a ∈ A such that ad exists. For interesting properties of the generalized Drazin inverse see [2, 4, 9–11]. For
a complete treatment of the generalized Drazin inverse, see [7, Ch. 2].
Let a ∈ A and let p ∈ A be an idempotent. We denote p = 1 − p. Then we can write
a = pap + pap + pap + pap.
Every idempotent p ∈ A induces a representation of an arbitrary element a ∈ A given by the following
matrix:
"
#
pap pap
.
a=
pap pap p
2010 Mathematics Subject Classification. Primary 46H05; Secondary 15A09
Keywords. Banach algebra, Generalized Drazin inverse, Additive result
Received: 10 June 2014; Accepted: 02 May 2015
Communicated by Dragan S. Djordjević
Email addresses: [email protected] (Xiaoji Liu), [email protected] (Xiaolan Qin), [email protected] (Julio Benı́tez)
X. Liu et al. / Filomat 30:8 (2016), 2289–2294
2290
Let a ∈ Ad and aπ = 1 − aad be the spectral idempotent of a corresponding to 0. It is well known that
a ∈ A can be represented in the following matrix form ([7, Ch. 2])
"
a1
0
a=
#
0
a2
,
(2)
p
where p = aad , a1 is invertible in the algebra pAp, ad is its inverse in pAp, and a2 is quasinilpotent in the
algebra pAp. Thus, the generalized Drazin inverse of a can be expressed as
"
ad =
ad
0
#
0
0
.
p
Obviously, if a ∈ Aqnil , then a is generalized Drazin invertible and ad = 0.
The motivation for this article is [3, 5, 6]. In these papers, the authors considered some conditions on
a, b ∈ A that allowed them to express (a + b)d in terms of a, ad , b, bd . Our aim in this paper is to investigate
the existence of the generalized Drazin inverse of the sum a + b and to give explicit expression for (a + b)d
under new conditions.
2. Main Results
A preliminary result witch will be used is the following:
Theorem 2.1. [3, Theorem 2.3] Let A be a Banach algebra, x, y ∈ A, and p ∈ A be an idempotent. Assume that x
and y are represented as
"
x=
a
c
#
0
b
"
,
y=
p
b
0
c
a
#
.
p
(i) If a ∈ (pAp)d and b ∈ (pAp)d , then x and y are generalized Drazin invertible, and
"
x =
d
ad
u
0
bd
#
"
,
y =
d
p
bd
0
u
ad
#
,
(3)
p
where
u=
∞
X
(bd )n+2 can aπ +
n=0
∞
X
bπ bn c(ad )n+2 − bd cad .
(4)
n=0
(ii) If x ∈ Ad and a ∈ (pAp)d , then b ∈ (pAp)d , and xd and yd are given by (3) and (4).
Theorem 2.2. [3, Corollary 3.4] Let A be a Banach algebra, b ∈ Ad , a ∈ Aqnil , and let ab = 0. Then a + b ∈ Ad and
(a + b)d =
∞
X
(bd )n+1 an .
n=0
The conditions aπ b = b and abaπ = 0 were used in [3, Theorem 4.1] to derive an expression of (a + b)d . In
Theorem 2.3, we will only use the condition abaπ = 0.
X. Liu et al. / Filomat 30:8 (2016), 2289–2294
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Theorem 2.3. Let A be a Banach algebra and let a, b ∈ Ad such that abaπ = 0 and aad baad ∈ Ad . Then a + b ∈ Ad
if and only if w = aad (a + b) ∈ Ad . In this case,
∞
X
(a + b)d = wd +
(bd )n+1 an aπ −
n=0
∞
X
(bd )n+1 an aπ bwd
n=0
∞

∞ X
∞
X
X


 (bd )n+k+2 ak  aπ bwn wπ + bπ
+
(a + b)n aπ b(wd )n+2


n=0
−
n=0
k=0
∞ X
∞
X
(bd )k+1 ak+1 (a + b)n aπ b(wd )n+2 .
n=0 k=0
Proof. Let p = aad . We can represent a as in (2), where a1 is invertible in the subalgebra pAp and a2 is
quasinilpotent. Hence,
"
#
ad 0
d
a =
.
(5)
0 0 p
Let us write
"
b1
b=
b3
b2
b4
#
.
(6)
p
From abaπ = 0 we have
"
# "
a1 0
b1
π
0 = aba =
0 a2 p b3
b2
b4
# "
p
0
0
0
aπ
#
"
=
p
0
0
a1 b2
a2 b4
#
.
p
Therefore, a1 b2 = 0 and a2 b4 = 0. Since a1 is invertible in pAp and b2 ∈ pA, we get b2 = 0. Hence
"
#
"
#
b1 0
a1 + b1
0
b=
,
a+b=
.
b3 b4 p
b3
a2 + b4 p
Observe that w = aad (a + b) = a1 + b1 .
Since b ∈ Ad and the hypothesis on b1 = aad baad , by Theorem 2.1 we get that b4 ∈ Ad . By using the
quasinilpotency of a2 and a2 b4 = 0, Theorem 2.2 leads to a2 + b4 ∈ Ad and
(a2 + b4 )d =
∞
X
(bd4 )n+1 an2 .
n=0
Thus, by Theorem 2.1, a + b is generalized Drazin invertible if and only if w = a1 + b1 is generalized Drazin
invertible. In this situation, we obtain
" d
#
w
0
d
(a + b) =
= wd + u + (a2 + b4 )d .
u (a2 + b4 )d p
and
u=
∞
X
((a2 + b4 )d )n+2 b3 wn wπ +
n=0
∞
X
(a2 + b4 )π (a2 + b4 )n b3 (wd )n+2 − (a2 + b4 )d b3 wd .
n=0
We have
(bd )n+1 an aπ =
"
(bd1 )n+1
∗
0
(bd4 )n+1
# "
p
an1
0
0
an2
# "
p
0
0
0
aπ
#
"
=
p
0
0
0
(bd4 )n+1 an2
#
= (bd4 )n+1 an2 .
p
X. Liu et al. / Filomat 30:8 (2016), 2289–2294
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Also,
∞
X
d n+1 n π
a a bw =
(b )
d
n=0
∞
X
(bd4 )n+1 an2 bwd = (a2 + b4 )d bwd
n=0
"
=
0
0
0
(a2 + b4 )d
# "
p
b1
b3
0
b4
# "
p
wd
0
0
0
#
= (a2 + b4 )d b3 wd .
p
In a similar way, we get
aπ bwn wπ = b3 wn wπ .
(7)
Now, we will find an expression for (a2 + b4 )π . To this end, we use a2 b4 = 0. Let us recall that a2 , b4 are
elements in the subalgebra pAp, where p = 1 − p = 1 − aad = aπ .
h
i
(a2 + b4 )π = aπ − (a2 + b4 )(a2 + b4 )d = aπ − (a2 + b4 ) bd4 + (bd4 )2 a2 + (bd4 )3 a22 + · · ·
i
i
h
h
= aπ − b4 bd4 + b4 (bd4 )2 a2 + b4 (bd4 )3 a22 + · · · = bπ4 − bd4 a2 + (bd4 )2 a22 + · · · ,
and so,
∞
X
(a2 + b4 )π (a2 + b4 )n b3 (wd )n+2
n=0
= bπ4
∞
X
(a2 + b4 )n b3 (wd )n+2 −
n=0
∞ X
∞
X
(bd4 )k+1 a2k+1 (a2 + b4 )n b3 (wd )n+2 .
n=0 k=0
One gets
(a2 + b4 )n b3 (wd )n+2 = (a + b)n aπ b(wd )n+2
and
n
d n+2
= (bd )k+1 ak+1 (a + b)n aπ b(wd )n+2 .
(bd4 )k+1 ak+1
2 (a2 + b4 ) b3 (w )
Finally, let us observe that the expression
(a2 + b4 )
d
n+2
=
∞
X
P
n+2
∞
d k+1 k
a
k=0 (b )
can be simplified. In effect, since
(bd4 )n+k+2 ak2 ,
k=0
we have that
∞
n+2
∞
X
X

 (bd )k+1 ak 
=
(bd )n+k+2 ak aπ .


k=0
k=0
The proof is finished.
If A is a Banach algebra, then we can define another multiplication in A by a b = ba. It is trivial that
(A, ) is a Banach algebra. If we apply Theorem 2.3 to this new algebra, we can immediately establish the
following result.
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Theorem 2.4. Let A be a Banach algebra and let a, b ∈ Ad such that aπ ba = 0 and aπ baπ ∈ Ad . Then a + b is
generalized Drazin invertible if and only if v = (a + b)aad is generalized Drazin invertible. In this case,
∞
X
(a + b) = v +
d
d
π n
d n+1
a a (b )
−
n=0
+
∞
X
n=0
−
∞
X
vd baπ an (bd )n+1
n=0
∞
 ∞
X
 X
π n π
k
d
n+k+2

 +
v v ba 
a (b )
(vd )n+2 baπ (a + b)n bπ

n=0
k=0
∞ X
∞
X
(vd )n+2 baπ (a + b)n ak+1 (bd )k+1 .
n=0 k=0
The condition aπ b = 0 is less general than aπ ba = 0. But if a, b ∈ A satisfy aπ b = 0, then the expression
for (a + b)d is simpler than the preceding theorems,
Theorem 2.5. Let A be a Banach algebra and let a, b ∈ Ad be such aπ b = 0. If w = aad (a + b) ∈ Ad , then a + b ∈ Ad
and
∞
X
(a + b)d = wd aad +
(wd )n+2 ban aπ .
n=0
If v = (a + b)aa ∈ A , then a + b ∈ Ad and
d
d
(a + b)d = vd +
∞
X
(vd )n+2 ban aπ .
n=0
Proof. Let us consider the matrix representations of a, ad , and b given in (2), (5), and (6) relative to the
idempotent p = aad . We will use the condition aπ b = 0. Since
# "
#
"
#
"
#
"
b1 b2
0 0
0 0
0 0
π
=
=
,
a b=
0 p p b3 b4 p
b3 b4 p
0 0 p
we obtain b3 = b4 = 0. Hence we have
"
#
a1 + b1 b2
a+b=
0
a2 p
and
"
w = aad (a + b) =
p
0
0
0
# "
p
a 1 + b1
0
b2
a2
"
#
=
p
a1 + b1
0
b2
0
#
.
(8)
p
Assume that w ∈ Ad . By Theorem 2.1, it follows that (a + b)d exists and
#
"
∞
X
(a1 + b1 )d u
d
(a + b) =
and
u=
((a1 + b1 )d )n+2 b2 an2 .
0
0 p
(9)
n=0
From (8) we have wd aad = (a1 + b1 )d and
  "
 (a1 + b1 )d n+2 ∗ 
d n+2 n π


(w ) ba a = 

0
0 p

 0 (a1 + b1 )d n+2 b2 an

2
= 
0
0
n+2
= (a1 + b1 )d
b2 an2 .
b1
0




p
b2
0
# "
p
an1
0
0
an2
# "
p
0
0
0
1−p
#
p
X. Liu et al. / Filomat 30:8 (2016), 2289–2294
2294
Hence the first part of the theorem follows. To prove the second part, observe that
"
#
"
# "
#
a 1 + b1 0
a1 + b1 b2
p 0
d
= a1 + b1
=
v = (a + b)aa =
0
a2 p 0 0 p
0
0 p
and (vd )n+2 ban aπ = ((a1 + b1 )d )n+2 b2 an2 . Now, the second part of the theorem can be proved by using (9).
As we have commented before, we can obtain a paired result by considering the Banach algebra A with
the product a b = ba. The key hypothesis of this new result will be baπ = 0.
Theorem 2.6. Let A be a Banach algebra and let a, b ∈ Ad be such baπ = 0. If v = (a + b)aad ∈ Ad , then a + b ∈ Ad
and
(a + b)d = aad vd +
∞
X
aπ an b(vd )n+2 .
n=0
If w = (a + b)aad ∈ Ad , then a + b ∈ Ad and
(a + b)d = wd +
∞
X
aπ an b(wd )n+2 .
n=0
Acknowledgments
The authors would thank the referee. His/her review has been very useful to improve the quality of the
manuscript. This work was supported by the National Natural Science Foundation of China (11361009).
References
[1] F.F. Bonsall, J. Duncan, Complete Normed Algebras, Springer Verlag, New York, 1973.
[2] N. Castro-González, J.J. Koliha, Additive perturbation results for the Drazin inverse, Linear Algebra and its Applications 397
(2005) 279–297.
[3] N. Castro-González, J.J. Koliha, New additive results for the Drazin inverse, Proceedings of the Royal Society of Edinburgh,
Section A 134 (2004) 1085–1097.
[4] N. Castro-González, J.J. Koliha, V. Rakočević, Continuity and general perturbation of the Drazin inverse for closed linear
operators, Abstract and Applied Analysis 7 (2002) 335–347.
[5] N. Castro-González, M.F. Martı́nez-Serrano, Expressions for the 1-Drazin inverse of additive perturbed elements in a Banach
algebra, Linear Algebra and its Applications 432 (2010) 1885–1895.
[6] D. S. Cvetković-Ilić, D.S. Djordjević, Y. Wei, Additive results for the generalized Drazin inverse in a Banach algebra, Linear
Algebra and its Applications 418 (2006) 53–61.
[7] D. Djordjević, V. Rakočević, Lectures on Generalized inverses, University of Niš, Niš, 2008.
[8] J.J. Koliha, A generalized Drazin inverse, Glasgow Mathematical Journal (1991) 367–381.
[9] J.J. Koliha, V. Rakočević, Differentiability of the g-Drazin inverse, Studia Mathematica 168 (2005) 193–201.
[10] J.J. Koliha, V. Rakočević, Holomorphic and meromorphic properties of the g-Drazin inverse, Demonstratio Mathematica 38 (2005)
657–666.
[11] X. Liu, L. Xu, Y.M. Yu, The representations of the Drazin inverse of differences of two matrices, Applied Mathematics and
Computation 216 (2010) 3652–3661.