vol21p85-97. 150KB Jun 04 2011 12:06:11 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 21, pp. 85-97, October 2010
ELA
http://math.technion.ac.il/iic/ela
PERTURBATION OF THE GENERALIZED DRAZIN INVERSE∗
CHUNYUAN DENG† AND YIMIN WEI‡
Abstract. In this paper, we investigate the perturbation of the generalized Drazin invertible
matrices and derive explicit generalized Drazin inverse expressions for the perturbations under certain
restrictions on the perturbing matrices.
Key words. generalized Drazin inverse, perturbation, index, block matrix.
AMS subject classifications. 47A05, 15A09.
1. Introduction. The Drazin inverse of an operator or a matrix has various applications in singular differential equations and singular difference equations, Markov
chains, and iterative methods (see [1, 2, 3, 4, 16, 17, 22, 23, 30]). In addition, the
perturbation analysis of the Drazin inverse is important from the perspectives of both
pure and computational mathematics (see [2]-[4],[6]-[7],[19]-[29]). The perturbation
results of the Drazin inverse of a closed linear operator can be applied to the solution of perturbed linear equations and to the asymptotic behavior of C0 -semigroup of
bounded linear operators [2, 16].
Let X denote an arbitrary complex Banach space and B(X) denote the Banach
algebra of all linear bounded operators on X. If A ∈ B(X), then R(A), N (A), σ(A)
and γ(A) denote the range, the kernel, the spectrum and the spectral radius of A,
respectively. Recall that an element B ∈ B(X) is the Drazin inverse of A ∈ B(X)
provided that
Ak+1 B = Ak ,
BAB = B
and
AB = BA
hold for some nonnegative integer k. The smallest k in the definition is called the
Drazin index of A, and is denoted by i(A). If A has the Drazin inverse, then the Drazin
∗ Received by the editors on June 10, 2009. Accepted for publication on July 31, 2010. Handling
Editors: Roger A. Horn and Fuzhen Zhang.
† School of Mathematics Science, South China Normal University, Guangzhou 510631, P. R. China
(cydeng@scnu.edu.cn). Supported by a grant from the Ph.D. Programs Foundation of Ministry of
Education of China (No. 20094407120001 ).
‡ Corresponding author: Y. Wei (ymwei@fudan.edu.cn and yimin.wei@gmail.com). School of
Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan
University, Shanghai, 200433, P. R. China. Supported by the National Natural Science Foundation
of China under grant 10871051, Shanghai Municipal Science & Technology Committee under grant
09DZ2272900, Shanghai Municipal Education Committee (Dawn Project), Doctoral Program of the
Ministry of Education under grant 20090071110003 and KLMM0901.
85
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 21, pp. 85-97, October 2010
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86
C. Deng and Y. Wei
inverse of A is unique and is denoted by AD . It is well-known that A ∈ B(X) has the
Drazin inverse if and only if the point λ = 0 is a pole of the resolvent λ 7→ (λI − A)−1 .
The order of this pole is equal to i(A). Particularly, it follows that 0 is not the
accumulation point of the spectrum σ(A) (see [1, 13, 17, 18]). For a compact subset
M of C, we use acc σ(M) to denote the set of all points of accumulation of M.
In [18], Koliha introduced the concept of a generalized Drazin inverse (or GDinverse). The GD-inverse of an element A ∈ B(X) exists if and only if 0 ∈
/ acc σ(A).
If 0 ∈
/ acc σ(A), then there exist open subsets U and V of C, such that σ(A)\{0} ⊂ U ,
0 ∈ V and U ∩ V = ∅. Define a function f in the following way:
0, λ ∈ V,
f (λ) =
1
λ ∈ U.
λ,
The function f is regular in a neighborhood of σ(A), i.e., it is analytic and singlevalued throughout a neighborhood of σ(A). The GD-inverse of A is defined as Ad =
f (A). If A is GD-invertible, then the spectral idempotent Aπ of A corresponding
to {0} is defined by Aπ = I − AAd . The matrix form of A with respect to the
decomposition X = N (Aπ ) ⊕ R(Aπ ) is given by
A = A1 ⊕ A2 ,
(1.1)
where A1 is invertible and A2 is quasinilpotent. If we denote CA = A1 ⊕ O and
QA = O ⊕ A2 , then A = CA + QA is known as the core-quasinilpotent decomposition
of A.
Let
U=
U11
O
U12
U22
and
B =A+U =
A1 + U11
O
U12
A2 + U22
with respect to the decomposition X = N (Aπ ) ⊕ R(Aπ ). In our paper, we consider
the perturbation of the generalized Drazin inverses in more generalized cases as follows:
• A1 + U11 is invertible and dim[R(Aπ )] is finite.
• A1 + U11 is invertible and U22 A2 = O.
• A1 + U11 is invertible and U22 A2 = A2 U22 .
In the existing literature [2, 12, 19, 24, 25, 26, 27, 28, 29], only are the following cases
considered:
• kI + A−1
1 U11 k < 1, which ensures that A1 + U11 is invertible, A2 + U22 is
quasinilpotent and U12 = O.
Electronic Journal of Linear Algebra ISSN 1081-3810
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87
• A1 +U11 is invertible, U22 is quasinilpotent with A2 U22 = U22 A2 and U12 = O.
• U11 = O, U22 is quasinilpotent and U22 A2 = O.
• kI + A1−1 U11 k < 1, U12 = O and U22 = O.
The main purpose of our paper is to study the perturbation
of the GD-inverse
Ad of a
d
d
bounded linear operator A and to obtain a bound for
A + U ) − A
under certain
conditions on the perturbing operator U , usually of small perturbation. The error
bounds for the perturbed GD-inverse without restriction on the perturbing matrix
(or operator) are difficult to obtain. We derive explicit expressions of GD-inverse for
the perturbations under certain restrictions on the perturbing operators.
2. Main results and proofs. To prove the main results, we begin with some
lemmas.
AB
Lemma 2.1. [15, Corollary 8] Let O
be a bounded linear operator on X ⊕ Y .
C
If σ(A) ∩ σ(C) has no-interior point, then
σ
A
O
B
C
= σ(A) ∪ σ(C).
For µ ∈ C and K ⊂ C, we define dist(µ, K) = inf{|λ − µ| : λ ∈ K} if K 6= ∅
and dist(µ, ∅) = ∞. The following result establishes the spectral characterization of
perturbation operators.
Lemma 2.2. If A and U ∈ B(X), B = A+U and σǫ (A) = {λ : dist(λ, σ(A)) < ǫ},
then for every ǫ > 0 there exists δ > 0 such that σ(B) ⊂ σǫ (A) whenever k U k< δ.
Proof. First we can suppose that δ = 1. Then it is clear that σ(A) and σ(B) are
in the closed disk D = {λ : |λ| ≤ 1+ k B k}. For every ǫ, D\σǫ (A) is a closed subset
in complex plane. If µ ∈ D\σǫ (A), then µI − A is invertible. So there exists an open
ball U(µI − A, δµ ) ⊂ B(X) such that T is invertible for all T ∈ U(µI − A, δµ ).
Let Eµ = {λ : |λ − µ| < 12 δµ }. Then {Eµ : µ ∈ D\σǫ (A)} is an open covering
of the compact subset D\σǫ (A). So there is a finite subcovering {Eµi , i = 1, 2, . . . , n}
covering the subset D\σǫ (A). Put δ = min{1, 12 δµi , i = 1, 2, . . . , n}. For every λ ∈
D\σǫ (A), if k U k< δ, then there exists an open subset Eµi so that λ ∈ Eµi . From
k λI − B − (µi I − A) k≤k A − B k +|λ − µi | < δ + 12 δµi < δµi , we have λI − B ∈
/ σ(B) and therefore σ(B) ⊂ σǫ (A).
U(µi I − A, δµi ). It follows that λ ∈
If A ∈ B(X) and γ(A) > 0, Koliha and Rakoˇcevi´c got the following basic lemma
[17, Lemma 1.3].
Lemma 2.3. [17, Lemma 1.3] Let A ∈ B(X) be Drazin invertible. If γ(A) > 0,
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
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C. Deng and Y. Wei
then
−1
dist(0, σ(A) \ {0}) = γ(AD )
.
Additive perturbation results for the GD-inverse were investigated by many authors (see [5, 7, 8, 12, 13, 16]). In [12], a detailed treatment of the generalized Drazin
inverse (A + B)d in terms of Ad and B d in infinite dimensional spaces was given.
Lemma 2.4. (1) [12, Theorem 2.3] Let A and B ∈ B(X) be GD-invertible with
AB = O. Then A + B is GD-invertible and
"∞
#
"∞
#
X
X
(A + B)d = (I − BB d )
B n (Ad )n Ad + B d
(B d )n An (I − AAd ).
n=0
n=0
(2) [12, Theorem 2.1] Let A, B, A + B ∈ B(X) be GD-invertible and AB = BA.
Then A + B is GD-invertible and
(A + B)d = (CA + CB )d [I + (CA + CB )d (QA + QB )]−1 ,
where A = CA + QA and B = CB + QB are known as the core-quasinilpotent decomposition of A and B, respectively.
Lemma 2.5. Let X be infinite dimensional Banach space. If A ∈ B(X) is a
nilpotent operator, then dim[N (A)] = ∞.
Proof. If A = O, it is obvious that dim[N (A)] = ∞. If A is non-zero nilpotent,
then there exists a positive integer k ≥ 2 such that Ak = O and Ak−1 6= O. Suppose
to the contrary that dim[N (A)] = n0 < ∞, then dim[R(A)] = ∞. It is clear that
dim[N (A) ∩ R(A)] ≤ n0 , which shows that dim[R(A2 )] = ∞. By induction, it is easy
to see that dim[R(Ak−1 )] = ∞ and R(Ak−1 ) ⊆ N (A). This is a contradiction.
For the Drazin index, if A and B are quasinilpotent, then i(A) and i(B) ∈ N ∪ ∞.
The following results were first obtained by Gonz´alez and Koliha for the case that A
and B are commuting quasinilpotent elements (see [2]). In fact, if both i(A) and i(B)
are finite or exactly one of i(A) and i(B) is finite (∞ − ∞ is not defined), then we
have the following generalization.
Lemma 2.6. Let A, B be quasinilpotent. If AB ∈ {O, ±A2 , ±B 2 , ±BA}, then
|i(A) − i(B)| + 1 ≤ i(A + B) ≤ i(A) + i(B) − 1
holds provided i(A) > 0 and the difference i(A) − i(B) is defined.
Proof. It is easy to obtain
(A + B)m =
m
X
i=0
B i Am−i if AB = O
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Perturbation of Drazin inverse
and
m
(A + B)m =
1 m X m−i−1 i m−i
2
BA
if AB = A2 ,
B +
2
i=0
we have i(A + B) ≤ i(A) + i(B) − 1. Similarly we have i(A + B) ≤ i(A) + i(B) − 1 if
AB ∈ {−A2 , ±B 2 , ±BA}.
If AB ∈ {O, ±A2 , ±B 2 , ±BA}, in a similar way to the proof of the first step, we
have i(A) = i((A + B) + (−B)) ≤ i(A + B) + i(B) − 1, i(B) = i((−A) + (A + B)) ≤
i(A + B) + i(B) − 1. So we obtain |i(A) − i(B)| + 1 ≤ i(A + B) whenever the difference
i(A) − i(B) is defined.
The following result has been proved in [12].
Lemma
and C ∈ B(Y, X),
2.7. [12] If A ∈ B(X) and B ∈ B(Y
) dare GD-invertible
AC
A X
d
then M = O B is GD-invertible and M = O B d , where
X=
"
∞
X
d n+2
(A )
n=0
CB
n
#
π
π
B +A
"
∞
X
n
d n+2
A C(B )
n=0
#
− Ad CB d .
First we discuss the GD-invertibility of the finite rank perturbation.
Theorem 2.8. Let A ∈ B(X) be GD-invertible, U ∈ B(X) satisfy Aπ U (I −Aπ ) =
O. There exists a δ > 0 such that B = A + U is Drazin invertible for each k U k< δ
if and only if dim[R(Aπ )] is finite.
Proof. “Necessity.” Let P = Aπ be the spectral idempotent of A corresponding
to {0}. Then A has the matrix form
(2.1)
A = A1 ⊕ A2
with respect to the decomposition X = N (P ) ⊕ R(P ), where A1 is invertible and A2
is quasinilpotent. If dim[R(Aπ )] is infinite, by Lemma 2.5, dim[N (A2 )] = ∞. Then
A2 has the following operator matrix form
O A23
A2 =
O A33
with respect to the space decomposition R(P ) = N (A2 ) ⊕ (R(P ) ⊖ N (A2 )), where
A33 is quasinilpotent. If S is a unilateral shift defined on N (A2 ), then σ(S) is a unit
1
disk. Define U by U = O ⊕ 2δ
S ⊕ O. Then k U k< δ and
A1
O
O
1
A+U = O
S A23
2δ
O
O
A33
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C. Deng and Y. Wei
with respect to the space decomposition X = N (P ) ⊕ N (A2 ) ⊕ R(P ) ⊖ N (A2 ) .
1
Thus σ(A + U ) = σ(A1 ) ∪ σ( 2δ
S) by Lemma 2.1. This shows that 0 is not an isolated
point of σ(A + U ). Therefore B = A + U is not Drazin invertible.
“Sufficiency.” If A is invertible, then the result holds. If A is quasinilpotent, then
dim(X) is finite because dim[R(Aπ )] is finite. Since all operators in B(X) are Drazin
invertible when dim(X) is finite, the result is obvious.
Suppose that A ∈ B(X) is GD-invertible, neither invertible nor quasinilpotent. If
A has the operator matrix form (2.1), then σ(A) = σ(A1 ) ∪ {0}, and σ(A1 ) ∩ {0} = ∅,
where σ(A2 ) = {0}. By Lemma 2.3, we have
−1
dist(0, σ(A) \ {0}) = γ(Ad )
> 0.
For small enough ǫ > 0, there exists two disjoint closed subsets M1 and M2 such
that σ ǫ (A1 ) = {λ : dist(λ, σ(A1 )) < ǫ} and σ ǫ (A2 ) = {λ : dist(λ, σ(A2 )) < ǫ} are
contained in the interior of M1 and M2 , respectively.
Since Aπ U (I − Aπ ) = O, U and B have the following operator matrix forms
U11 U12
A1 + U11
U12
U=
, B=
O U22
O
A2 + U22
with respect to the decomposition X = N (Aπ ) ⊕ R(Aπ ). By Lemma 2.2, there is
a δ > 0 such that σ(A1 + U11 ) ⊂ σ ǫ (A1 ) ⊂ M1 and σ(A2 + U22 ) ⊂ σ ǫ (A2 ) ⊂ M2
whenever k U k< δ. Thus σ(B) = σ(A1 + U11 ) ∪ σ(A2 + U22 ) by Lemma 2.1. Since
dim[R(Aπ )] is finite, σ(A2 +U22 ) is a finite subset of σ(B). Note that 0 ∈
/ σ(A1 +U11 ).
So 0 is not an accumulated point of σ(B). Therefore B = A + U is Drazin invertible.
Definition 2.9. Let A ∈ B(X) be GD-invertible and δ > 0.
(i) An operator U ∈ B(X) is called Aδ −compatible if it satisfies Aπ U (I −Aπ ) = O
and if Aπ U is GD-invertible for each k U k< δ.
(ii) If the operator U is Aδ −compatible and satisfies AAπ U = O, then we say U
is A0δ −compatible.
(iii) If the operator U is Aδ −compatible and satisfies AAπ U = Aπ U A, and −1
is not the accumulation point of σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then we say U is
A−1
δ −compatible.
Next we consider the perturbation of the generalized Drazin inverse. We need
some notation. Let
νA = (I + Ad U )−1 Ad ,
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Perturbation of Drazin inverse
d
d
−1
κA = CA
,
π U (I + CAπ U A)
!
∞
X
π
d
π
d n
π
n
ωA = (A U )
(A U )
(A A)
.
n=0
Theorem 2.10. Let A ∈ B(X) be GD-invertible. Then there always exists a
δ > 0 such that A + U is GD-invertible whenever U ∈ B(X) is A0δ −compatible. In
this case, we have
#
"∞
X
n+2
d
π
π
n
(A + U ) = νA + ωA − νA U ωA +
νA U A (A (A + U )) × I − (A + U )ωA
n=0
and the inequality
|i(A) − i(Aπ U )| ≤ i(A + U ) ≤ i(A) + i(Aπ U ) − 1
holds provided i(A) > 0 and the difference i(A) − i(Aπ U ) is defined.
Proof. Let B = A+U and P = Aπ be the spectral idempotent of A corresponding
to {0}. If U is A0δ −compatible, then A, U and B have the matrix forms
(2.2) A =
A1
O
O
A2
;
U=
U11
O
U12
U22
;
B=
A1 + U11
O
U12
A2 + U22
with respect to the decomposition X = N (P ) ⊕ R(P ), where A1 is invertible, A2 is
quasinilpotent, U22 is GD-invertible and A2 U22 = O.
If A is invertible, then i(A) = 0 and Aπ = O. By Lemma 2.1, there exists a δ > 0
such that B is invertible, B −1 = (I + A−1 U )−1 A−1 and i(B) = 0. The results hold.
Similarly, by Lemma 2.4 and Lemma 2.6, we can show that the results hold if A is
quasinilpotent.
Now suppose that A ∈ B(X) is GD-invertible with i(A) > 0 and σ(A) 6= {0}.
For small enough ǫ > 0, there are two disjoint closed subsets M1 and M2 such that
σ ǫ (A1 ) ⊂ M1 and σ ǫ (A2 ) ⊂ M2 . By Lemma 2.2, there exists a δ > 0 such that
σ(A1 + U11 ) ⊂ σ ǫ (A1 ) and σ(A2 + U22 ) ⊂ σ ǫ (A2 ) whenever k U k< δ (this implies
that k U11 k< δ and k U22 k< δ). So we have σ(A1 + U11 ) ∩ σ(A2 + U22 ) = ∅. It
follows that σ(B) = σ(A1 + U11 ) ∪ σ(A2 + U22 ) by Lemma 2.1. Hence, there always
exists a δ > 0 such that A1 + U11 is invertible whenever k U k< δ.
Since Aπ U is GD-invertible and AAπ U = O, A2 + U22 is GD-invertible and
#
"∞
X
d
d
d n n
(A2 + U22 ) = U22
(U22 ) A2
n=0
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C. Deng and Y. Wei
by Lemma 2.4. Hence, there exists a δ > 0 such that A1 +U11 is invertible and A2 +U22
is GD-invertible whenever k U k< δ. It follows that B = A + U is GD-invertible. By
Lemma 2.7, we have
(A1 + U11 )−1
S
d
,
B =
O
(A2 + U22 )d
where
S=
∞
X
(A1 + U11 )−(n+2) U12 (A2 + U22 )n ](A2 + U22 )π − (A1 + U11 )−1 U12 (A2 + U22 )d .
n=0
Note that (A1 + U11 )−1 ⊕ O = (I + Ad U )−1 Ad = νA ,
π
d
d
[A (A + U )] = O ⊕ (A2 + U22 ) = O ⊕
π
d
= (A U )
"
∞
X
π
(A U )
n=0
= ωA
d n
d
U22
"
∞
X
d n n
(U22
) A2
n=0
π
n
(A A)
#
#
and
O
O
S
O
=
"
∞
X
n+2
U Aπ (Aπ (A
νA
#
+ U ))n × [I − (A + U )ωA ] − νA U ωA .
n=0
So we get
d
B = νA + ω A − νA U ω A +
"
∞
X
n+2
νA
U Aπ (Aπ (A
n=0
n
+ U ))
#
× I − (A + U )ωA .
Note that A = A1 ⊕ A2 and A1 + U11 is invertible. Then i(A) = i(A2 ) and
i(B) = i(A2 + U22 ). Since A2 is quasinilpotent, U22 is GD-invertible and A2 U22 = O,
we have
′
U22 O
O A′2
,
U22 =
A2 =
′′
O U22
O A′′2
π
π
′
with respect to the decomposition R(Aπ ) = N (U22
)⊕R(U22
), where U22
is invertible,
′′
′′
′′ ′′
A2 and U22 are quasinilpotent with A2 U22 = O. By Lemma 2.6,
′′
′′
′′
|i(A′′2 ) − i(U22
)| + 1 ≤ i(A′′2 + U22
) ≤ i(A′′2 ) + i(U22
) − 1.
From i(A′′2 ) + 1 ≥ i(A2 ) ≥ i(A′′2 ), we have
′′
′′
′′
|i(A2 ) − i(U22
)| ≤ i(A′′2 + U22
) ≤ i(A2 ) + i(U22
) − 1.
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93
′′
′′
From i(B) = i(A2 + U22 ) = i(A′′2 + U22
), i(A2 ) = i(A) and i(U22
) = i(U22 ) = i(Aπ U ),
we have
|i(A) − i(Aπ U )| ≤ i(B) ≤ i(A) + i(Aπ U ) − 1.
The following result is an extension of [2, Theorem 2.3].
Theorem 2.11. Let A ∈ B(X) be GD-invertible. Then there always exists a
δ > 0 such that A + U is GD-invertible whenever U ∈ B(X) is A−1
δ -compatible. In
this case,
#
"∞
X
n+2
d
π
π
n
(A + U ) = νA + κA − νA U κA +
νA U A (A (A + U )) × I − (A + U )κA ,
n=0
where CAπ U is invertible component in the core-quasinilpotent decomposition Aπ U =
CAπ U + QAπ U . In particular, if −1 ∈
/ σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then
|i((Aπ U )π A) − i(Aπ U )| + 1 ≤ i(A + U ) ≤ i(A) + i(Aπ U ) − 1
holds, provided that i(A) > 0 and the difference i[(Aπ U )π A] − i(Aπ U ) is defined.
π
Proof. Let B = A + U . Since U is A−1
δ -compatible, A U is GD-invertible and
π
A U has the core-quasinilpotent decomposition form A U = CAπ U + QAπ U . If A, U
and B have the representations as (2.2), then
π
Aπ U = O ⊕ U22
and
A2 U22 = U22 A2 .
Let P1 be the spectral idempotent of U22 in the Banach algebra B R(Aπ ) cor-
2
1
⊕ U22
with respect to the
responding to {0}. Then U22 has the form U22 = U22
π
1
space decomposition R(A ) = N (P1 ) ⊕ R(P1 ), where U22 is invertible with CAπ U =
2
2
1
.
is quasinilpotent with QAπ U = O ⊕ O ⊕ U22
O ⊕ U22
⊕ O, U22
Since A2 commutes with P1 , we have A2 = A12 ⊕ A22 with respect to the space
decomposition R(Aπ ) = N (P1 ) ⊕ R(P1 ), where Ai2 (i = 1, 2) are quasinilpotent and
2
i
i
+ A22 is quasinilpotent and
, (i = 1, 2). So U22
Ai2 = Ai2 U22
U22
1 −1 1
(Aπ U )d Aπ U (I + (Aπ U )d A) = O ⊕ (I + (U22
) A2 ) ⊕ O.
If −1 ∈
/ σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then
1
2
O ⊕ (A2 + U22 ) = O ⊕ (U22
+ A12 ) ⊕ (U22
+ A22 )
is GD-invertible. If −1 ∈ σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then by the definition of
1 −1 1
A−1
A2 is GD-invertible.
δ −compatible, −1 is the isolate point and so I + (U22 )
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1
1 −1
1
Note that A12 , U22
and (U22
) commute pairwise, U22
+ A12 is GD-invertible and
1
1 −1
1 −1 1 d
1 −1 1 d
1 −1
(U22
+ A12 )d = (U22
) (I + (U22
) A2 ) = (I + (U22
) A2 ) (U22
) . Hence
2
1
+ A12 ) ⊕ (U22
+ A22 )
O ⊕ (A2 + U22 ) = O ⊕ (U22
is GD-invertible. By Lemma 2.4, we have
−1
d
d
(Aπ (A + U ))d = (O ⊕ (A2 + U22 ))d = CA
= κA .
π U (I + CAπ U A)
Hence, by Lemma 2.2 and Lemma 2.3, there exists δ > 0, if k U k< δ, then A1 + U11 is
invertible and A2 + U22 is GD-invertible. It follows that B = A + U is GD-invertible.
By Lemma 2.7, we have
Bd =
where
"
S=
∞
X
(A1 + U11 )−1
O
S
(A2 + U22 )d
,
#
(A1 + U11 )−(n+2) U12 (A2 + U22 )n ](A2 + U22 )π − (A1 + U11 )−1 U12 (A2 + U22 )d .
n=0
Note that (A1 + U11 )−1 ⊕ O = (I + Ad U )−1 Ad = νA and
#
"X
∞
O S
n+2
π
π
n
=
νA U A (A (A + U )) × I − (A + U )κA − νA U κA .
O O
n=0
So we obtain
d
B = νA + κ A − νA U κ A +
"
∞
X
n=0
n+2
νA
U Aπ (Aπ (A
n
+ U ))
#
× I − (A + U )κA .
From (2.2) we know i(A) = i(A2 ), i[(Aπ U )π A] = i(A22 ) and i(B) = i(A2 + U22 ).
2
Since −1 ∈
/ σ((Aπ U )D Aπ U (I + (Aπ U )d A)), i(A2 + U22 ) = i(A22 + U22
) and i(A2 ) ≥
2
2
2
2
2
2
2
i(A2 ). By Lemma 2.6, |i(A2 ) − i(U22 )| + 1 ≤ i(A2 + U22 ) ≤ i(A2 ) + i(U22
) − 1. Hence
|i((Aπ U )π A) − i(Aπ U )| + 1 ≤ i(B) ≤ i(A) + i(Aπ U ) − 1.
3. Special cases. Many interesting special cases of Theorem 2.10 and Theorem
2.11 were considered in [2, 23], Some of them are generalizations of the well-known
results. Wei and Wang [23] considered a perturbation of the Drazin inverse of A under
the conditions on U given by
(3.1)
AAD U AAD = U and k AD U k< 1.
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Perturbation of Drazin inverse
If A and U have the forms as (2.2), then condition (3.1) implies that Ui2 = O (i = 1, 2).
If δ is small enough, then the following result is a direct corollary of Theorem 2.10
and 2.11.
Corollary 3.1. [23, Theorem 3.2] Suppose that B = A+U with U = AAd U AAd
and k Ad U k< 1 hold. Then
B d − Ad = −B d U Ad = −Ad U B d ,
B d = (I + Ad U )−1 Ad = Ad (I + U Ad )−1 ,
k B d − Ad k
k Ad U k
≤
.
k Ad k
1− k Ad U k
Castro Gonz´
alez and Koliha [2] considered a perturbation of the Drazin inverse
of A under the A-compatible conditions on U given by
(3.2)
Aπ U = U Aπ ,
σ(Aπ U ) = {0}
and
AAπ U = U AAπ
and stated that U is inverse-A-compatible (alternatively norm-A-compatible) if U is
A-compatible and satisfies I + Ad U invertible (alternatively k Ad U k< 1). Let A and
U have the forms in (2.2). It is clear that if U is inverse-A-compatible, then U is
Aδ−1 −compatible with additional conditions A1 U11 = U11 A1 , U12 = O, I + A1−1 U11
invertible and U22 quasinilpotent. These imply that CAπ U = O and i((Aπ U )π A) =
i(A). Then Theorem 2.3 in [2] is recovered as a special case of Theorem 2.11.
Corollary 3.2. [2, Theorem 2.3] Let A ∈ B(X) be GD-invertibIe and U ∈ B(X)
be inverse-A-compatible. If B = A + U , then B is GD-invertible, and
N (B π ) = N (Aπ ) and R(B π ) = R(Aπ ),
B d = (I − Ad U )−1 Ad = Ad (I − U Ad )−1
and
|i(A) − i(Aπ U )| + 1 ≤ i(B) ≤ i(A) + i(Aπ U ) − 1
holds, provided that i(A) > 0 and the difference i(A) − i(Aπ U ) is defined.
Let A, U and B have the forms in (2.2). In fact, the cases treated in [2] and
[23] allow the perturbation of the invertible component of A = A1 ⊕ A2 , and the
quasinilpotent perturbation of the quasinilpotent component of A = A1 ⊕ A2 . In
Theorem 2.10 and Theorem 2.11, if δ is small enough, then k U11 k< δ and
−1
k Ad U k=k A−1
1 U11 ⊕ O k=k A1 k δ < 1.
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C. Deng and Y. Wei
So the following result is a direct corollary.
Corollary 3.3. [2, Corollary 2.5] Let A ∈ B(X) be GD-invertible and B = A+U
with U ∈ B(X) norm-A-compatible. Then
k B d − Ad k
k Ad U k
≤
d
kA k
1− k Ad U k
and
k Ad k
k Ad k
d
k≤
≤k
B
.
1+ k Ad U k
1− k Ad U k
4. Concluding remarks. In this paper, we investigate the perturbation of the
generalized Drazin invertible operators and derive explicit generalized Drazin inverse
expressions for the perturbations under certain restrictions on the perturbing operators. It is natural to ask if we can extend our results to the W-weighted Drazin
inverse [6, 10, 21], which will be the future research topic.
Acknowledgments. The authors would like to thank referees for their very
detailed comments which greatly improve the presentation of our paper.
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A publication of the International Linear Algebra Society
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PERTURBATION OF THE GENERALIZED DRAZIN INVERSE∗
CHUNYUAN DENG† AND YIMIN WEI‡
Abstract. In this paper, we investigate the perturbation of the generalized Drazin invertible
matrices and derive explicit generalized Drazin inverse expressions for the perturbations under certain
restrictions on the perturbing matrices.
Key words. generalized Drazin inverse, perturbation, index, block matrix.
AMS subject classifications. 47A05, 15A09.
1. Introduction. The Drazin inverse of an operator or a matrix has various applications in singular differential equations and singular difference equations, Markov
chains, and iterative methods (see [1, 2, 3, 4, 16, 17, 22, 23, 30]). In addition, the
perturbation analysis of the Drazin inverse is important from the perspectives of both
pure and computational mathematics (see [2]-[4],[6]-[7],[19]-[29]). The perturbation
results of the Drazin inverse of a closed linear operator can be applied to the solution of perturbed linear equations and to the asymptotic behavior of C0 -semigroup of
bounded linear operators [2, 16].
Let X denote an arbitrary complex Banach space and B(X) denote the Banach
algebra of all linear bounded operators on X. If A ∈ B(X), then R(A), N (A), σ(A)
and γ(A) denote the range, the kernel, the spectrum and the spectral radius of A,
respectively. Recall that an element B ∈ B(X) is the Drazin inverse of A ∈ B(X)
provided that
Ak+1 B = Ak ,
BAB = B
and
AB = BA
hold for some nonnegative integer k. The smallest k in the definition is called the
Drazin index of A, and is denoted by i(A). If A has the Drazin inverse, then the Drazin
∗ Received by the editors on June 10, 2009. Accepted for publication on July 31, 2010. Handling
Editors: Roger A. Horn and Fuzhen Zhang.
† School of Mathematics Science, South China Normal University, Guangzhou 510631, P. R. China
(cydeng@scnu.edu.cn). Supported by a grant from the Ph.D. Programs Foundation of Ministry of
Education of China (No. 20094407120001 ).
‡ Corresponding author: Y. Wei (ymwei@fudan.edu.cn and yimin.wei@gmail.com). School of
Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan
University, Shanghai, 200433, P. R. China. Supported by the National Natural Science Foundation
of China under grant 10871051, Shanghai Municipal Science & Technology Committee under grant
09DZ2272900, Shanghai Municipal Education Committee (Dawn Project), Doctoral Program of the
Ministry of Education under grant 20090071110003 and KLMM0901.
85
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C. Deng and Y. Wei
inverse of A is unique and is denoted by AD . It is well-known that A ∈ B(X) has the
Drazin inverse if and only if the point λ = 0 is a pole of the resolvent λ 7→ (λI − A)−1 .
The order of this pole is equal to i(A). Particularly, it follows that 0 is not the
accumulation point of the spectrum σ(A) (see [1, 13, 17, 18]). For a compact subset
M of C, we use acc σ(M) to denote the set of all points of accumulation of M.
In [18], Koliha introduced the concept of a generalized Drazin inverse (or GDinverse). The GD-inverse of an element A ∈ B(X) exists if and only if 0 ∈
/ acc σ(A).
If 0 ∈
/ acc σ(A), then there exist open subsets U and V of C, such that σ(A)\{0} ⊂ U ,
0 ∈ V and U ∩ V = ∅. Define a function f in the following way:
0, λ ∈ V,
f (λ) =
1
λ ∈ U.
λ,
The function f is regular in a neighborhood of σ(A), i.e., it is analytic and singlevalued throughout a neighborhood of σ(A). The GD-inverse of A is defined as Ad =
f (A). If A is GD-invertible, then the spectral idempotent Aπ of A corresponding
to {0} is defined by Aπ = I − AAd . The matrix form of A with respect to the
decomposition X = N (Aπ ) ⊕ R(Aπ ) is given by
A = A1 ⊕ A2 ,
(1.1)
where A1 is invertible and A2 is quasinilpotent. If we denote CA = A1 ⊕ O and
QA = O ⊕ A2 , then A = CA + QA is known as the core-quasinilpotent decomposition
of A.
Let
U=
U11
O
U12
U22
and
B =A+U =
A1 + U11
O
U12
A2 + U22
with respect to the decomposition X = N (Aπ ) ⊕ R(Aπ ). In our paper, we consider
the perturbation of the generalized Drazin inverses in more generalized cases as follows:
• A1 + U11 is invertible and dim[R(Aπ )] is finite.
• A1 + U11 is invertible and U22 A2 = O.
• A1 + U11 is invertible and U22 A2 = A2 U22 .
In the existing literature [2, 12, 19, 24, 25, 26, 27, 28, 29], only are the following cases
considered:
• kI + A−1
1 U11 k < 1, which ensures that A1 + U11 is invertible, A2 + U22 is
quasinilpotent and U12 = O.
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87
• A1 +U11 is invertible, U22 is quasinilpotent with A2 U22 = U22 A2 and U12 = O.
• U11 = O, U22 is quasinilpotent and U22 A2 = O.
• kI + A1−1 U11 k < 1, U12 = O and U22 = O.
The main purpose of our paper is to study the perturbation
of the GD-inverse
Ad of a
d
d
bounded linear operator A and to obtain a bound for
A + U ) − A
under certain
conditions on the perturbing operator U , usually of small perturbation. The error
bounds for the perturbed GD-inverse without restriction on the perturbing matrix
(or operator) are difficult to obtain. We derive explicit expressions of GD-inverse for
the perturbations under certain restrictions on the perturbing operators.
2. Main results and proofs. To prove the main results, we begin with some
lemmas.
AB
Lemma 2.1. [15, Corollary 8] Let O
be a bounded linear operator on X ⊕ Y .
C
If σ(A) ∩ σ(C) has no-interior point, then
σ
A
O
B
C
= σ(A) ∪ σ(C).
For µ ∈ C and K ⊂ C, we define dist(µ, K) = inf{|λ − µ| : λ ∈ K} if K 6= ∅
and dist(µ, ∅) = ∞. The following result establishes the spectral characterization of
perturbation operators.
Lemma 2.2. If A and U ∈ B(X), B = A+U and σǫ (A) = {λ : dist(λ, σ(A)) < ǫ},
then for every ǫ > 0 there exists δ > 0 such that σ(B) ⊂ σǫ (A) whenever k U k< δ.
Proof. First we can suppose that δ = 1. Then it is clear that σ(A) and σ(B) are
in the closed disk D = {λ : |λ| ≤ 1+ k B k}. For every ǫ, D\σǫ (A) is a closed subset
in complex plane. If µ ∈ D\σǫ (A), then µI − A is invertible. So there exists an open
ball U(µI − A, δµ ) ⊂ B(X) such that T is invertible for all T ∈ U(µI − A, δµ ).
Let Eµ = {λ : |λ − µ| < 12 δµ }. Then {Eµ : µ ∈ D\σǫ (A)} is an open covering
of the compact subset D\σǫ (A). So there is a finite subcovering {Eµi , i = 1, 2, . . . , n}
covering the subset D\σǫ (A). Put δ = min{1, 12 δµi , i = 1, 2, . . . , n}. For every λ ∈
D\σǫ (A), if k U k< δ, then there exists an open subset Eµi so that λ ∈ Eµi . From
k λI − B − (µi I − A) k≤k A − B k +|λ − µi | < δ + 12 δµi < δµi , we have λI − B ∈
/ σ(B) and therefore σ(B) ⊂ σǫ (A).
U(µi I − A, δµi ). It follows that λ ∈
If A ∈ B(X) and γ(A) > 0, Koliha and Rakoˇcevi´c got the following basic lemma
[17, Lemma 1.3].
Lemma 2.3. [17, Lemma 1.3] Let A ∈ B(X) be Drazin invertible. If γ(A) > 0,
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C. Deng and Y. Wei
then
−1
dist(0, σ(A) \ {0}) = γ(AD )
.
Additive perturbation results for the GD-inverse were investigated by many authors (see [5, 7, 8, 12, 13, 16]). In [12], a detailed treatment of the generalized Drazin
inverse (A + B)d in terms of Ad and B d in infinite dimensional spaces was given.
Lemma 2.4. (1) [12, Theorem 2.3] Let A and B ∈ B(X) be GD-invertible with
AB = O. Then A + B is GD-invertible and
"∞
#
"∞
#
X
X
(A + B)d = (I − BB d )
B n (Ad )n Ad + B d
(B d )n An (I − AAd ).
n=0
n=0
(2) [12, Theorem 2.1] Let A, B, A + B ∈ B(X) be GD-invertible and AB = BA.
Then A + B is GD-invertible and
(A + B)d = (CA + CB )d [I + (CA + CB )d (QA + QB )]−1 ,
where A = CA + QA and B = CB + QB are known as the core-quasinilpotent decomposition of A and B, respectively.
Lemma 2.5. Let X be infinite dimensional Banach space. If A ∈ B(X) is a
nilpotent operator, then dim[N (A)] = ∞.
Proof. If A = O, it is obvious that dim[N (A)] = ∞. If A is non-zero nilpotent,
then there exists a positive integer k ≥ 2 such that Ak = O and Ak−1 6= O. Suppose
to the contrary that dim[N (A)] = n0 < ∞, then dim[R(A)] = ∞. It is clear that
dim[N (A) ∩ R(A)] ≤ n0 , which shows that dim[R(A2 )] = ∞. By induction, it is easy
to see that dim[R(Ak−1 )] = ∞ and R(Ak−1 ) ⊆ N (A). This is a contradiction.
For the Drazin index, if A and B are quasinilpotent, then i(A) and i(B) ∈ N ∪ ∞.
The following results were first obtained by Gonz´alez and Koliha for the case that A
and B are commuting quasinilpotent elements (see [2]). In fact, if both i(A) and i(B)
are finite or exactly one of i(A) and i(B) is finite (∞ − ∞ is not defined), then we
have the following generalization.
Lemma 2.6. Let A, B be quasinilpotent. If AB ∈ {O, ±A2 , ±B 2 , ±BA}, then
|i(A) − i(B)| + 1 ≤ i(A + B) ≤ i(A) + i(B) − 1
holds provided i(A) > 0 and the difference i(A) − i(B) is defined.
Proof. It is easy to obtain
(A + B)m =
m
X
i=0
B i Am−i if AB = O
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Perturbation of Drazin inverse
and
m
(A + B)m =
1 m X m−i−1 i m−i
2
BA
if AB = A2 ,
B +
2
i=0
we have i(A + B) ≤ i(A) + i(B) − 1. Similarly we have i(A + B) ≤ i(A) + i(B) − 1 if
AB ∈ {−A2 , ±B 2 , ±BA}.
If AB ∈ {O, ±A2 , ±B 2 , ±BA}, in a similar way to the proof of the first step, we
have i(A) = i((A + B) + (−B)) ≤ i(A + B) + i(B) − 1, i(B) = i((−A) + (A + B)) ≤
i(A + B) + i(B) − 1. So we obtain |i(A) − i(B)| + 1 ≤ i(A + B) whenever the difference
i(A) − i(B) is defined.
The following result has been proved in [12].
Lemma
and C ∈ B(Y, X),
2.7. [12] If A ∈ B(X) and B ∈ B(Y
) dare GD-invertible
AC
A X
d
then M = O B is GD-invertible and M = O B d , where
X=
"
∞
X
d n+2
(A )
n=0
CB
n
#
π
π
B +A
"
∞
X
n
d n+2
A C(B )
n=0
#
− Ad CB d .
First we discuss the GD-invertibility of the finite rank perturbation.
Theorem 2.8. Let A ∈ B(X) be GD-invertible, U ∈ B(X) satisfy Aπ U (I −Aπ ) =
O. There exists a δ > 0 such that B = A + U is Drazin invertible for each k U k< δ
if and only if dim[R(Aπ )] is finite.
Proof. “Necessity.” Let P = Aπ be the spectral idempotent of A corresponding
to {0}. Then A has the matrix form
(2.1)
A = A1 ⊕ A2
with respect to the decomposition X = N (P ) ⊕ R(P ), where A1 is invertible and A2
is quasinilpotent. If dim[R(Aπ )] is infinite, by Lemma 2.5, dim[N (A2 )] = ∞. Then
A2 has the following operator matrix form
O A23
A2 =
O A33
with respect to the space decomposition R(P ) = N (A2 ) ⊕ (R(P ) ⊖ N (A2 )), where
A33 is quasinilpotent. If S is a unilateral shift defined on N (A2 ), then σ(S) is a unit
1
disk. Define U by U = O ⊕ 2δ
S ⊕ O. Then k U k< δ and
A1
O
O
1
A+U = O
S A23
2δ
O
O
A33
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with respect to the space decomposition X = N (P ) ⊕ N (A2 ) ⊕ R(P ) ⊖ N (A2 ) .
1
Thus σ(A + U ) = σ(A1 ) ∪ σ( 2δ
S) by Lemma 2.1. This shows that 0 is not an isolated
point of σ(A + U ). Therefore B = A + U is not Drazin invertible.
“Sufficiency.” If A is invertible, then the result holds. If A is quasinilpotent, then
dim(X) is finite because dim[R(Aπ )] is finite. Since all operators in B(X) are Drazin
invertible when dim(X) is finite, the result is obvious.
Suppose that A ∈ B(X) is GD-invertible, neither invertible nor quasinilpotent. If
A has the operator matrix form (2.1), then σ(A) = σ(A1 ) ∪ {0}, and σ(A1 ) ∩ {0} = ∅,
where σ(A2 ) = {0}. By Lemma 2.3, we have
−1
dist(0, σ(A) \ {0}) = γ(Ad )
> 0.
For small enough ǫ > 0, there exists two disjoint closed subsets M1 and M2 such
that σ ǫ (A1 ) = {λ : dist(λ, σ(A1 )) < ǫ} and σ ǫ (A2 ) = {λ : dist(λ, σ(A2 )) < ǫ} are
contained in the interior of M1 and M2 , respectively.
Since Aπ U (I − Aπ ) = O, U and B have the following operator matrix forms
U11 U12
A1 + U11
U12
U=
, B=
O U22
O
A2 + U22
with respect to the decomposition X = N (Aπ ) ⊕ R(Aπ ). By Lemma 2.2, there is
a δ > 0 such that σ(A1 + U11 ) ⊂ σ ǫ (A1 ) ⊂ M1 and σ(A2 + U22 ) ⊂ σ ǫ (A2 ) ⊂ M2
whenever k U k< δ. Thus σ(B) = σ(A1 + U11 ) ∪ σ(A2 + U22 ) by Lemma 2.1. Since
dim[R(Aπ )] is finite, σ(A2 +U22 ) is a finite subset of σ(B). Note that 0 ∈
/ σ(A1 +U11 ).
So 0 is not an accumulated point of σ(B). Therefore B = A + U is Drazin invertible.
Definition 2.9. Let A ∈ B(X) be GD-invertible and δ > 0.
(i) An operator U ∈ B(X) is called Aδ −compatible if it satisfies Aπ U (I −Aπ ) = O
and if Aπ U is GD-invertible for each k U k< δ.
(ii) If the operator U is Aδ −compatible and satisfies AAπ U = O, then we say U
is A0δ −compatible.
(iii) If the operator U is Aδ −compatible and satisfies AAπ U = Aπ U A, and −1
is not the accumulation point of σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then we say U is
A−1
δ −compatible.
Next we consider the perturbation of the generalized Drazin inverse. We need
some notation. Let
νA = (I + Ad U )−1 Ad ,
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d
d
−1
κA = CA
,
π U (I + CAπ U A)
!
∞
X
π
d
π
d n
π
n
ωA = (A U )
(A U )
(A A)
.
n=0
Theorem 2.10. Let A ∈ B(X) be GD-invertible. Then there always exists a
δ > 0 such that A + U is GD-invertible whenever U ∈ B(X) is A0δ −compatible. In
this case, we have
#
"∞
X
n+2
d
π
π
n
(A + U ) = νA + ωA − νA U ωA +
νA U A (A (A + U )) × I − (A + U )ωA
n=0
and the inequality
|i(A) − i(Aπ U )| ≤ i(A + U ) ≤ i(A) + i(Aπ U ) − 1
holds provided i(A) > 0 and the difference i(A) − i(Aπ U ) is defined.
Proof. Let B = A+U and P = Aπ be the spectral idempotent of A corresponding
to {0}. If U is A0δ −compatible, then A, U and B have the matrix forms
(2.2) A =
A1
O
O
A2
;
U=
U11
O
U12
U22
;
B=
A1 + U11
O
U12
A2 + U22
with respect to the decomposition X = N (P ) ⊕ R(P ), where A1 is invertible, A2 is
quasinilpotent, U22 is GD-invertible and A2 U22 = O.
If A is invertible, then i(A) = 0 and Aπ = O. By Lemma 2.1, there exists a δ > 0
such that B is invertible, B −1 = (I + A−1 U )−1 A−1 and i(B) = 0. The results hold.
Similarly, by Lemma 2.4 and Lemma 2.6, we can show that the results hold if A is
quasinilpotent.
Now suppose that A ∈ B(X) is GD-invertible with i(A) > 0 and σ(A) 6= {0}.
For small enough ǫ > 0, there are two disjoint closed subsets M1 and M2 such that
σ ǫ (A1 ) ⊂ M1 and σ ǫ (A2 ) ⊂ M2 . By Lemma 2.2, there exists a δ > 0 such that
σ(A1 + U11 ) ⊂ σ ǫ (A1 ) and σ(A2 + U22 ) ⊂ σ ǫ (A2 ) whenever k U k< δ (this implies
that k U11 k< δ and k U22 k< δ). So we have σ(A1 + U11 ) ∩ σ(A2 + U22 ) = ∅. It
follows that σ(B) = σ(A1 + U11 ) ∪ σ(A2 + U22 ) by Lemma 2.1. Hence, there always
exists a δ > 0 such that A1 + U11 is invertible whenever k U k< δ.
Since Aπ U is GD-invertible and AAπ U = O, A2 + U22 is GD-invertible and
#
"∞
X
d
d
d n n
(A2 + U22 ) = U22
(U22 ) A2
n=0
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C. Deng and Y. Wei
by Lemma 2.4. Hence, there exists a δ > 0 such that A1 +U11 is invertible and A2 +U22
is GD-invertible whenever k U k< δ. It follows that B = A + U is GD-invertible. By
Lemma 2.7, we have
(A1 + U11 )−1
S
d
,
B =
O
(A2 + U22 )d
where
S=
∞
X
(A1 + U11 )−(n+2) U12 (A2 + U22 )n ](A2 + U22 )π − (A1 + U11 )−1 U12 (A2 + U22 )d .
n=0
Note that (A1 + U11 )−1 ⊕ O = (I + Ad U )−1 Ad = νA ,
π
d
d
[A (A + U )] = O ⊕ (A2 + U22 ) = O ⊕
π
d
= (A U )
"
∞
X
π
(A U )
n=0
= ωA
d n
d
U22
"
∞
X
d n n
(U22
) A2
n=0
π
n
(A A)
#
#
and
O
O
S
O
=
"
∞
X
n+2
U Aπ (Aπ (A
νA
#
+ U ))n × [I − (A + U )ωA ] − νA U ωA .
n=0
So we get
d
B = νA + ω A − νA U ω A +
"
∞
X
n+2
νA
U Aπ (Aπ (A
n=0
n
+ U ))
#
× I − (A + U )ωA .
Note that A = A1 ⊕ A2 and A1 + U11 is invertible. Then i(A) = i(A2 ) and
i(B) = i(A2 + U22 ). Since A2 is quasinilpotent, U22 is GD-invertible and A2 U22 = O,
we have
′
U22 O
O A′2
,
U22 =
A2 =
′′
O U22
O A′′2
π
π
′
with respect to the decomposition R(Aπ ) = N (U22
)⊕R(U22
), where U22
is invertible,
′′
′′
′′ ′′
A2 and U22 are quasinilpotent with A2 U22 = O. By Lemma 2.6,
′′
′′
′′
|i(A′′2 ) − i(U22
)| + 1 ≤ i(A′′2 + U22
) ≤ i(A′′2 ) + i(U22
) − 1.
From i(A′′2 ) + 1 ≥ i(A2 ) ≥ i(A′′2 ), we have
′′
′′
′′
|i(A2 ) − i(U22
)| ≤ i(A′′2 + U22
) ≤ i(A2 ) + i(U22
) − 1.
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93
′′
′′
From i(B) = i(A2 + U22 ) = i(A′′2 + U22
), i(A2 ) = i(A) and i(U22
) = i(U22 ) = i(Aπ U ),
we have
|i(A) − i(Aπ U )| ≤ i(B) ≤ i(A) + i(Aπ U ) − 1.
The following result is an extension of [2, Theorem 2.3].
Theorem 2.11. Let A ∈ B(X) be GD-invertible. Then there always exists a
δ > 0 such that A + U is GD-invertible whenever U ∈ B(X) is A−1
δ -compatible. In
this case,
#
"∞
X
n+2
d
π
π
n
(A + U ) = νA + κA − νA U κA +
νA U A (A (A + U )) × I − (A + U )κA ,
n=0
where CAπ U is invertible component in the core-quasinilpotent decomposition Aπ U =
CAπ U + QAπ U . In particular, if −1 ∈
/ σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then
|i((Aπ U )π A) − i(Aπ U )| + 1 ≤ i(A + U ) ≤ i(A) + i(Aπ U ) − 1
holds, provided that i(A) > 0 and the difference i[(Aπ U )π A] − i(Aπ U ) is defined.
π
Proof. Let B = A + U . Since U is A−1
δ -compatible, A U is GD-invertible and
π
A U has the core-quasinilpotent decomposition form A U = CAπ U + QAπ U . If A, U
and B have the representations as (2.2), then
π
Aπ U = O ⊕ U22
and
A2 U22 = U22 A2 .
Let P1 be the spectral idempotent of U22 in the Banach algebra B R(Aπ ) cor-
2
1
⊕ U22
with respect to the
responding to {0}. Then U22 has the form U22 = U22
π
1
space decomposition R(A ) = N (P1 ) ⊕ R(P1 ), where U22 is invertible with CAπ U =
2
2
1
.
is quasinilpotent with QAπ U = O ⊕ O ⊕ U22
O ⊕ U22
⊕ O, U22
Since A2 commutes with P1 , we have A2 = A12 ⊕ A22 with respect to the space
decomposition R(Aπ ) = N (P1 ) ⊕ R(P1 ), where Ai2 (i = 1, 2) are quasinilpotent and
2
i
i
+ A22 is quasinilpotent and
, (i = 1, 2). So U22
Ai2 = Ai2 U22
U22
1 −1 1
(Aπ U )d Aπ U (I + (Aπ U )d A) = O ⊕ (I + (U22
) A2 ) ⊕ O.
If −1 ∈
/ σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then
1
2
O ⊕ (A2 + U22 ) = O ⊕ (U22
+ A12 ) ⊕ (U22
+ A22 )
is GD-invertible. If −1 ∈ σ[(Aπ U )d Aπ U (I + (Aπ U )d A)], then by the definition of
1 −1 1
A−1
A2 is GD-invertible.
δ −compatible, −1 is the isolate point and so I + (U22 )
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1
1 −1
1
Note that A12 , U22
and (U22
) commute pairwise, U22
+ A12 is GD-invertible and
1
1 −1
1 −1 1 d
1 −1 1 d
1 −1
(U22
+ A12 )d = (U22
) (I + (U22
) A2 ) = (I + (U22
) A2 ) (U22
) . Hence
2
1
+ A12 ) ⊕ (U22
+ A22 )
O ⊕ (A2 + U22 ) = O ⊕ (U22
is GD-invertible. By Lemma 2.4, we have
−1
d
d
(Aπ (A + U ))d = (O ⊕ (A2 + U22 ))d = CA
= κA .
π U (I + CAπ U A)
Hence, by Lemma 2.2 and Lemma 2.3, there exists δ > 0, if k U k< δ, then A1 + U11 is
invertible and A2 + U22 is GD-invertible. It follows that B = A + U is GD-invertible.
By Lemma 2.7, we have
Bd =
where
"
S=
∞
X
(A1 + U11 )−1
O
S
(A2 + U22 )d
,
#
(A1 + U11 )−(n+2) U12 (A2 + U22 )n ](A2 + U22 )π − (A1 + U11 )−1 U12 (A2 + U22 )d .
n=0
Note that (A1 + U11 )−1 ⊕ O = (I + Ad U )−1 Ad = νA and
#
"X
∞
O S
n+2
π
π
n
=
νA U A (A (A + U )) × I − (A + U )κA − νA U κA .
O O
n=0
So we obtain
d
B = νA + κ A − νA U κ A +
"
∞
X
n=0
n+2
νA
U Aπ (Aπ (A
n
+ U ))
#
× I − (A + U )κA .
From (2.2) we know i(A) = i(A2 ), i[(Aπ U )π A] = i(A22 ) and i(B) = i(A2 + U22 ).
2
Since −1 ∈
/ σ((Aπ U )D Aπ U (I + (Aπ U )d A)), i(A2 + U22 ) = i(A22 + U22
) and i(A2 ) ≥
2
2
2
2
2
2
2
i(A2 ). By Lemma 2.6, |i(A2 ) − i(U22 )| + 1 ≤ i(A2 + U22 ) ≤ i(A2 ) + i(U22
) − 1. Hence
|i((Aπ U )π A) − i(Aπ U )| + 1 ≤ i(B) ≤ i(A) + i(Aπ U ) − 1.
3. Special cases. Many interesting special cases of Theorem 2.10 and Theorem
2.11 were considered in [2, 23], Some of them are generalizations of the well-known
results. Wei and Wang [23] considered a perturbation of the Drazin inverse of A under
the conditions on U given by
(3.1)
AAD U AAD = U and k AD U k< 1.
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Perturbation of Drazin inverse
If A and U have the forms as (2.2), then condition (3.1) implies that Ui2 = O (i = 1, 2).
If δ is small enough, then the following result is a direct corollary of Theorem 2.10
and 2.11.
Corollary 3.1. [23, Theorem 3.2] Suppose that B = A+U with U = AAd U AAd
and k Ad U k< 1 hold. Then
B d − Ad = −B d U Ad = −Ad U B d ,
B d = (I + Ad U )−1 Ad = Ad (I + U Ad )−1 ,
k B d − Ad k
k Ad U k
≤
.
k Ad k
1− k Ad U k
Castro Gonz´
alez and Koliha [2] considered a perturbation of the Drazin inverse
of A under the A-compatible conditions on U given by
(3.2)
Aπ U = U Aπ ,
σ(Aπ U ) = {0}
and
AAπ U = U AAπ
and stated that U is inverse-A-compatible (alternatively norm-A-compatible) if U is
A-compatible and satisfies I + Ad U invertible (alternatively k Ad U k< 1). Let A and
U have the forms in (2.2). It is clear that if U is inverse-A-compatible, then U is
Aδ−1 −compatible with additional conditions A1 U11 = U11 A1 , U12 = O, I + A1−1 U11
invertible and U22 quasinilpotent. These imply that CAπ U = O and i((Aπ U )π A) =
i(A). Then Theorem 2.3 in [2] is recovered as a special case of Theorem 2.11.
Corollary 3.2. [2, Theorem 2.3] Let A ∈ B(X) be GD-invertibIe and U ∈ B(X)
be inverse-A-compatible. If B = A + U , then B is GD-invertible, and
N (B π ) = N (Aπ ) and R(B π ) = R(Aπ ),
B d = (I − Ad U )−1 Ad = Ad (I − U Ad )−1
and
|i(A) − i(Aπ U )| + 1 ≤ i(B) ≤ i(A) + i(Aπ U ) − 1
holds, provided that i(A) > 0 and the difference i(A) − i(Aπ U ) is defined.
Let A, U and B have the forms in (2.2). In fact, the cases treated in [2] and
[23] allow the perturbation of the invertible component of A = A1 ⊕ A2 , and the
quasinilpotent perturbation of the quasinilpotent component of A = A1 ⊕ A2 . In
Theorem 2.10 and Theorem 2.11, if δ is small enough, then k U11 k< δ and
−1
k Ad U k=k A−1
1 U11 ⊕ O k=k A1 k δ < 1.
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C. Deng and Y. Wei
So the following result is a direct corollary.
Corollary 3.3. [2, Corollary 2.5] Let A ∈ B(X) be GD-invertible and B = A+U
with U ∈ B(X) norm-A-compatible. Then
k B d − Ad k
k Ad U k
≤
d
kA k
1− k Ad U k
and
k Ad k
k Ad k
d
k≤
≤k
B
.
1+ k Ad U k
1− k Ad U k
4. Concluding remarks. In this paper, we investigate the perturbation of the
generalized Drazin invertible operators and derive explicit generalized Drazin inverse
expressions for the perturbations under certain restrictions on the perturbing operators. It is natural to ask if we can extend our results to the W-weighted Drazin
inverse [6, 10, 21], which will be the future research topic.
Acknowledgments. The authors would like to thank referees for their very
detailed comments which greatly improve the presentation of our paper.
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