Representations for the generalized inverses of a modified operator
Some explicit representations for the generalized inverses of a modified operator $A + YGZ$ are derived under some conditions, where $A, Y, Z$, and $G$ are operators between Banach spaces. These results generalize the recent works about the Drazin inverse and the Moore – Penrose inverse of complex m...
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Institute of Mathematics, NAS of Ukraine
2016
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507769867599872 |
|---|---|
| author | Mosić, D. Мосич, Д. |
| author_facet | Mosić, D. Мосич, Д. |
| author_sort | Mosić, D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2019-12-05T09:30:37Z |
| description | Some explicit representations for the generalized inverses of a modified operator $A + YGZ$ are derived under some conditions, where $A, Y, Z$, and $G$ are operators between Banach spaces. These results generalize the recent works about
the Drazin inverse and the Moore – Penrose inverse of complex matrices and Hilbert-space operators. |
| first_indexed | 2026-03-24T02:14:35Z |
| format | Article |
| fulltext |
UDC 517.9
D. Mosić (Univ. Niš, Serbia)
REPRESENTATIONS FOR THE GENERALIZED INVERSES
OF A MODIFIED OPERATOR*
ЗОБРАЖЕННЯ УЗАГАЛЬНЕНИХ ОБЕРНЕНИХ ОПЕРАТОРIВ
ДЛЯ МОДИФIКОВАНОГО ОПЕРАТОРА
Some explicit representations for the generalized inverses of a modified operator A + Y GZ are derived under some
conditions, where A, Y, Z, and G are operators between Banach spaces. These results generalize the recent works about
the Drazin inverse and the Moore – Penrose inverse of complex matrices and Hilbert-space operators.
За певних умов встановлено деякi явнi зображення узагальнених обернених операторiв для модифiкованого опера-
тора A + Y GZ, де A, Y, Z та G — оператори мiж банаховими просторами. Цi результати узагальнюють останнi
роботи щодо матриць Дразiна та Мура – Пенроуза, обернених до комплексних матриць та операторiв у гiльбертовому
просторi.
1. Introduction. Let H, K and L be arbitrary Banach spaces. We use \scrB (H,K) to denote the set of
all linear bounded operators from H to K. Set \scrB (H) = \scrB (H,H). For an operator A \in \scrB (H,K), the
symbols N(A) and R(A) will denote the null space and the range of A, respectively. An operator
B \in \scrB (K,H), B \not = 0, is an outer inverse of A, if BAB = B holds. The outer inverse B of A
with the prescribed range T and the null space S is uniquely determined and it is denoted by A
(2)
T,S .
Some results concerning the outer invertible operators on Banach spaces are presented in [5].
An element a of a unital Banach algebra \scrA is said to be hermitian, if \| \mathrm{e}\mathrm{x}\mathrm{p} (ita)\| = 1, for all
t \in \BbbR . If \scrA is a C\ast -algebra, then a \in \scrA is Hermitian if and only if a is self-adjoint.
The Moore – Penrose inverse of A \in \scrB (H,K) is the operator B \in \scrB (K,H) which satisfies
(1) ABA = A, (2) BAB = B, (3) AB \mathrm{i}\mathrm{s} \mathrm{H}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{n}, (4) BA \mathrm{i}\mathrm{s} \mathrm{H}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{n}.
Recall that according to [7] (Lemma 2.1), there is at most one Moore – Penrose inverse and it is
denoted by A\dagger . If A is a linear bounded operator between two Hilbert spaces, then A\dagger exists if and
only if R(A) is closed (see [1]).
Let W \in \scrB (K,H) be a fixed nonzero operator. An operator A \in \scrB (H,K) is called Wg-Drazin
invertible, if there exists some B \in \scrB (H,K) satisfying
(5) BWAWB = B, (6) AWB = BWA, (7) A - AWBWA \mathrm{i}\mathrm{s} \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}.
The Wg-Drazin inverse B of A is unique, if it exists, and denoted by Ad,W [3]. If H = K,
A \in \scrB (H) and W = IH , then B = Ad is the generalized Drazin inverse, or Koliha – Drazin inverse
of A [6]. The Drazin inverse is a special case of the generalized Drazin inverse for which A - A2B
is nilpotent instead of A - A2B is quasinilpotent. Obviously, if A is Drazin invertible, then it
is generalized Drazin invertible. The group inverse is the Drazin inverse for which the condition
A - A2B is nilpotent is replaced with A = ABA. We use A\# to denote the group inverse of A.
Denote by (5) the equality AB = BA and by (6) the condition A - ABA is quasinilpotent.
* The author is supported by the Ministry of Science, Republic of Serbia (grant no. 174007).
c\bigcirc D. MOSIĆ, 2016
860 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
REPRESENTATIONS FOR THE GENERALIZED INVERSES OF A MODIFIED OPERATOR 861
If \delta \subset \{ 1, 2, 3, 4, 5, 6\} and B satisfies the equations (i) for all i \in \delta , then B is an \delta -inverse of
A. The set of all \delta -inverse of A is denote by A\{ \delta \} . Observe that A\{ 1, 2, 3, 4\} = \{ A\dagger \} , A\{ 2, 5, 6\} =
= \{ Ad\} and A\{ 1, 2, 5\} = \{ A\#\} .
The generalized inverses and its applications are very important in various applied mathematical
fields like numerical analysis, singular differential equations, singular difference equations, Markov
chains, etc.
The Sherman – Morrison – Woodbury formula (or SMW formula) [8, 10] related to the inverse of
matrix, i.e., the formula (A+ Y GZT ) - 1 = A - 1 - A - 1Y (G - 1 + ZTA - 1Y ) - 1ZTA - 1, is an useful
computational tool in applications to statistics, networks, structural analysis, asymptotic analysis,
optimization and partial differential equations. Deng [4] considered the more generalized case of
the SMW formula when A and A + Y GZ\ast are not invertible operator. Precisely, some conditions
under which the SMW formula can be represented in the Moore – Penrose inverse and the generalized
Drazin inverse forms for Hilbert space operators are investigated.
Wei [9] and J. Chen, Z. Xu [2] have discussed the expression of the Drazin inverse and the
weighted Drazin inverse of a modified square matrix A - CB. These results can be applied to update
finite Markov chains.
In this paper, we will prove that Sherman – Morrison – Woodbury formula has the analogous
result concerning the \delta -inverse of a modified operator A + Y GZ between Banach spaces, where
\delta \subset \{ 1, 2, 3, 4, 5\} . Under some conditions, we will show that the Wg-Drazin inverse of a modified
operator A+ Y GZ exists and can be represented in terms of generalized SMW forms if and only if
the Wg-Drazin inverse of the initial operator A exists. As a consequence, some results of Deng [4],
Wei [9] and J. Chen, Z. Xu [2] are obtained.
2. Results. In the first theorem of this section, we consider the generalized inverse of a
modified operator A+ Y GZ and, under certain circumstances, we get that A\prime \in A\{ \delta \} if and only if
A\prime - A\prime Y S\prime ZA\prime \in (A + Y GZ)\{ \delta \} (\delta \subset \{ 1, 2, 3, 4, 5\} ). Thus, the generalized inverse of operator
A + Y GZ is expressed in terms of generalized SMW formula. Notice that, we do not assume that
S\prime and G\prime are the generalized inverses of S and G, respectively, that is, S\prime and G\prime can be arbitrary
operators.
Theorem 2.1. Suppose that A \in \scrB (H,K), A\prime \in \scrB (K,H), Y \in \scrB (L,K), Z \in \scrB (H,L) and
G,G\prime , S\prime \in \scrB (L). Let B = A+ Y GZ and S = G\prime + ZA\prime Y.
(a) If \delta \subset \{ 1, 2, 4\} and
ZA\prime A = Z, Y S\prime SGZ = Y GZ, Y S\prime G\prime GZ = Y S\prime Z, (2.1)
then A\prime \in A\{ \delta \} if and only if A\prime - A\prime Y S\prime ZA\prime \in (A+ Y GZ)\{ \delta \} .
(b) If \delta \subset \{ 1, 2, 3\} and
AA\prime Y = Y, Y GZ = Y GSS\prime Z, Y S\prime Z = Y GG\prime S\prime Z, (2.2)
then A\prime \in A\{ \delta \} if and only if A\prime - A\prime Y S\prime ZA\prime \in (A+ Y GZ)\{ \delta \} .
(c) If the conditions (2.1) and (2.2) hold, then A\prime \in A\{ 5\} if and only if A\prime - A\prime Y S\prime ZA\prime \in
\in (A+ Y GZ)\{ 5\} .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
862 D. MOSIĆ
Proof. (a) If P = A\prime - A\prime Y S\prime ZA\prime , by (2.1), we obtain
PB = (A\prime - A\prime Y S\prime ZA\prime )(A+ Y GZ) =
= A\prime A+A\prime Y GZ - A\prime Y S\prime (ZA\prime A) - A\prime Y S\prime (ZA\prime Y )GZ =
= A\prime A+A\prime Y GZ - A\prime Y S\prime Z - A\prime Y S\prime (S - G\prime )GZ =
= A\prime A+A\prime Y (IL - S\prime S)GZ - A\prime Y S\prime (IL - G\prime G)Z = A\prime A.
So, A\prime \in A\{ 4\} if and only if P \in B\{ 4\} . Since, from AA\prime Y = Y,
PBP = A\prime AP = A\prime AA\prime - A\prime (AA\prime Y )S\prime ZA\prime = A\prime AA\prime - A\prime Y S\prime ZA\prime ,
we deduce that A\prime AA\prime = A\prime \leftrightarrow PBP = P, i.e., A\prime \in A\{ 2\} \leftrightarrow P \in B\{ 2\} . Further, by the equalities
BPB = (A+ Y GZ)A\prime A = AA\prime A+ Y G(ZA\prime A) = AA\prime A+ Y GZ,
A\prime \in A\{ 1\} is equivalent to P \in B\{ 1\} .
(b) The assumptions (2.2) imply
BP = (A+ Y GZ)(A\prime - A\prime Y S\prime ZA\prime ) =
= AA\prime - (AA\prime Y )S\prime ZA\prime + Y GZA\prime - Y G(ZA\prime Y )S\prime ZA\prime =
= AA\prime - Y S\prime ZA\prime - Y GZA\prime - Y G(S - G\prime )S\prime ZA\prime =
= AA\prime - Y (IL - GG\prime )S\prime ZA\prime - Y G(IL - SS\prime )ZA\prime = AA\prime .
Hence, A\prime \in A\{ 3\} if and only if P \in B\{ 3\} . The rest follows in the same way as in the proof of part
(a).
(c) Since PB = A\prime A and BP = AA\prime , we deduce that A\prime \in A\{ 5\} if and only if P \in B\{ 5\} .
Theorem 2.1 is proved.
As an application of Theorem 2.1, we get the following result related to the ordinary inverse, the
outer inverse and the Moore – Penrose inverse.
Corollary 2.1. Suppose that the conditions of Theorem 2.1 are satisfied.
(a) Let T and S be subspaces of H and K, respectively, and let the equalities (2.1) or (2.2) hold.
Then there exists A
(2)
T,S and A\prime = A
(2)
T,S if and only if (A + Y GZ)
(2)
T,S exists and (A + Y GZ)
(2)
T,S =
= A\prime - A\prime Y S\prime ZA\prime . Furthermore, if A(2)
T,S exists, then (A+ Y GZ)
(2)
T,S = A
(2)
T,S - A
(2)
T,SY S\prime ZA
(2)
T,S .
In addition, let the equalities (2.1) and (2.2) hold.
(b) Then there exists A\dagger and A\prime = A\dagger if and only if (A + Y GZ)\dagger exists and (A + Y GZ)\dagger =
= A\prime - A\prime Y S\prime ZA\prime . Furthermore, if A\dagger exists, then (A+ Y GZ)\dagger = A\dagger - A\dagger Y S\prime ZA\dagger .
(c) Then A is invertible and A\prime = A - 1 if and only if A+Y GZ is invertible and (A+Y GZ) - 1 =
= A\prime - A\prime Y S\prime ZA\prime . Furthermore, if A is invertible, then (A+ Y GZ) - 1 = A - 1 - A - 1Y S\prime ZA - 1.
Proof. (a) By Theorem 2.1, A\prime \in A\{ 2\} is equivalent to P = A\prime - A\prime Y S\prime ZA\prime \in B\{ 2\} . Observe
that R(P ) = R(PB) = R(A\prime A) = R(A\prime ) and N(P ) = N(BP ) = N(AA\prime ) = N(A\prime ). Thus,
A\prime = A
(2)
T,S \leftrightarrow P = B
(2)
T,S .
The statements (b) and (c) follows directly from Theorem 2.1.
In the following theorem, we consider the explicit expression for the Wg-Drazin inverse of a
modified operator A+ Y GZ.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
REPRESENTATIONS FOR THE GENERALIZED INVERSES OF A MODIFIED OPERATOR 863
Theorem 2.2. Suppose that A,A\prime \in \scrB (H,K), W \in \scrB (K,H), Y \in \scrB (L,K), Z \in \scrB (H,L)
and G,G\prime , S\prime \in \scrB (L). Let B = A+ Y GZ and S = G\prime + ZWA\prime WY. If
ZWA\prime WA = Z, Y S\prime SGZ = Y GZ = Y GSS\prime Z,
AWA\prime WY = Y, Y S\prime G\prime GZ = Y S\prime Z = Y GG\prime S\prime Z,
then there exists Ad,W and Ad,W = A\prime if and only if (A+Y GZ)d,W exists and (A+Y GZ)d,W = A\prime -
- A\prime WY S\prime ZWA\prime . Furthermore, if Ad,W exists, then (A+Y GZ)d,W=Ad,W - Ad,WWY S\prime ZWAd,W .
Proof. In the similar way as in the proof of Theorem 2.1, for T = A\prime - A\prime WY S\prime ZWA\prime , we
obtain TWB = A\prime WA and BWT = AWA\prime implying A\prime WA = AWA\prime \leftrightarrow TWB = BWT. From
TWBWT = A\prime WAWT, we get TWBWT = T is equivalent to A\prime WAWA\prime = A\prime . Also, we have
B - BWTWB = A+ Y GZ - AWA\prime W (A+ Y GZ) = A - AWA\prime WA,
which gives B - BWTWB is quasinilpotent if and only if A - AWA\prime WA is quasinilpotent. So,
Ad,W = A\prime \leftrightarrow Bd,W = T.
Theorem 2.2 is proved.
If H = K and W = IH in Theorem 2.2, we obtain the next corollary concerning representations
for the generalized Drazin inverse and the group inverse of a operator on Banach space.
Corollary 2.2. Let A,A\prime \in \scrB (H), Y \in \scrB (L,H), Z \in \scrB (H,L) and G,G\prime , S\prime \in \scrB (L). Set
B = A+ Y GZ and S = G\prime + ZA\prime Y. Assume that the equalities (2.1) and (2.2) hold.
(a) There exists Ad and A\prime = Ad if and only if (A + Y GZ)d exists and (A + Y GZ)d =
= A\prime - A\prime Y S\prime ZA\prime . Furthermore, if Ad exists, then (A+ Y GZ)d = Ad - AdY S\prime ZAd.
(b) There exists A\# and A\prime = A\# if and only if (A + Y GZ)\# exists and (A + Y GZ)\# =
= A\prime - A\prime Y S\prime ZA\prime . Furthermore, if A\# exists, then (A+ Y GZ)\# = A\# - A\#Y S\prime ZA\#.
(c) If \delta \subset \{ 1, 2, 3, 4, 5, 6\} , then A\prime \in A\{ \delta \} if and only if A\prime - A\prime Y S\prime ZA\prime \in (A+ Y GZ)\{ \delta \} .
Remark. In Theorem 2.1, Corollary 2.1, Theorem 2.2 and Corollary 2.2, if we assume that G\prime
is \theta -inverse of G and S\prime is \vargamma -inverse of S, for \theta , \vargamma \subset \{ 1 - 6\} , then we obtain various formulas
for corresponding inverse of A + Y GZ. In particular, if G\prime and S\prime are the Moore – Penrose (or the
generalized Drazin) inverses of G and S, respectively, in Corollary 2.1 (or Corollary 2.2), we get as
a spacial case [4] (Theorem 2.2) (or [4], Theorem 2.4) for Hilbert space operators.
For G = G\prime = IL in Theorems 2.1 and 2.2, we have the next results.
Corollary 2.3. Suppose that \delta \subset \{ 1, 2, 3, 4\} , A \in \scrB (H,K), A\prime \in \scrB (K,H), Y \in \scrB (L,K),
Z \in \scrB (H,L) and S\prime \in \scrB (L). Let B = A+ Y Z and S = IL + ZA\prime Y. If
ZA\prime A = Z, AA\prime Y = Y, Y S\prime SZ = Y Z = Y SS\prime Z,
then A\prime \in A\{ \delta \} if and only if A\prime - A\prime Y S\prime ZA\prime \in (A+ Y Z)\{ \delta \} .
Notice that [9] (Theorem 2.1) and [2] (Theorem 2.1) concerning the Drazin inverse and the
weighted Drazin inverse of complex matrices are particular cases of the following corollary.
Corollary 2.4. Suppose that A,A\prime \in \scrB (H,K), W \in \scrB (K,H), Y \in \scrB (L,K), Z \in \scrB (H,L)
and S\prime \in \scrB (L). Let B = A+ Y Z and S = IL + ZWA\prime WY. If
ZWA\prime WA = Z, AWA\prime WY = Y, Y S\prime SZ = Y Z = Y SS\prime Z,
then there exists Ad,W and Ad,W = A\prime if and only if (A+Y Z)d,W exists and (A+Y Z)d,W = A\prime -
- A\prime WY S\prime ZWA\prime . Furthermore, if Ad,W exists, then (A+Y Z)d,W = Ad,W - Ad,WWY S\prime ZWAd,W .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
864 D. MOSIĆ
If S is invertible and S\prime = S - 1 in Corollaries 2.3 and 2.4, the following consequences recover
[4] (Corollaries 2.3 and 2.6).
Corollary 2.5. Suppose that \delta \subset \{ 1, 2, 3, 4\} , A \in \scrB (H,K), A\prime \in \scrB (K,H), Y \in \scrB (L,K),
Z \in \scrB (H,L), S\prime \in \scrB (L) and B = A+ Y Z. Let S = IL + ZA\prime Y be invertible and S\prime = S - 1. If
ZA\prime A = Z, and AA\prime Y = Y,
then A\prime \in A\{ \delta \} if and only if A\prime - A\prime Y (I + ZA\prime Y ) - 1ZA\prime \in (A+ Y Z)\{ \delta \} .
Corollary 2.6. Suppose that A,A\prime \in \scrB (H,K), W \in \scrB (K,H), Y \in \scrB (L,K), Z \in \scrB (H,L),
S\prime \in \scrB (L) and B = A+ Y Z. Let S = IL + ZWA\prime WY be invertible and S\prime = S - 1. If
ZWA\prime WA = Z, AWA\prime WY = Y,
then there exists Ad,W and Ad,W = A\prime if and only if (A + Y Z)d,W exists and (A + Y Z)d,W =
= A\prime - A\prime WY (IL + ZWA\prime WY ) - 1ZWA\prime . Furthermore, if Ad,W exists, then (A + Y Z)d,W =
= Ad,W - Ad,WWY (IL + ZWA\prime WY ) - 1ZWAd,W .
References
1. Ben-Israel A., Greville T. N. E. Generalized inverses: theory and applications. – 2nd ed. – New York: Springer, 2003.
2. Chen J., Xu Z. Representations for the weighted Drazin inverse of a modified matrix // Appl. Math. and Comput. –
2008. – 203. – P. 202 – 209.
3. Dajić A., Koliha J. J. The weighted g-Drazin inverse for operators // J. Austral. Math. Soc. – 2007. – 82. – P. 163 – 181.
4. Deng C. Y. A generalization of the Sherman – Morrison – Woodbury formula // Appl. Math. Lett. – 2011. – 24. –
P. 1561 – 1564.
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6. Koliha J. J. A generalized Drazin inverse // Glasgow Math. J. – 1996. – 38. – P. 367 – 381.
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Received 08.07.14
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
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| id | umjimathkievua-article-1885 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:35Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/25/daebcb8ed8ac9417df0b30d9383f9d25.pdf |
| spelling | umjimathkievua-article-18852019-12-05T09:30:37Z Representations for the generalized inverses of a modified operator Зображення узагальнених обернених операторiв для модифiкованого оператора Mosić, D. Мосич, Д. Some explicit representations for the generalized inverses of a modified operator $A + YGZ$ are derived under some conditions, where $A, Y, Z$, and $G$ are operators between Banach spaces. These results generalize the recent works about the Drazin inverse and the Moore – Penrose inverse of complex matrices and Hilbert-space operators. За певних умов встановлено деякi явнi зображення узагальнених обернених операторiв для модифiкованого оператора $A + YGZ$, де $A, Y, Z$ та $G$ — оператори мiж банаховими просторами. Цi результати узагальнюють останнi роботи щодо матриць Дразiна та Мура – Пенроуза, обернених до комплексних матриць та операторiв у гiльбертовому просторi. Institute of Mathematics, NAS of Ukraine 2016-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1885 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 6 (2016); 860-864 Український математичний журнал; Том 68 № 6 (2016); 860-864 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1885/867 Copyright (c) 2016 Mosić D. |
| spellingShingle | Mosić, D. Мосич, Д. Representations for the generalized inverses of a modified operator |
| title | Representations for the generalized inverses of a modified operator |
| title_alt | Зображення узагальнених обернених операторiв для модифiкованого оператора |
| title_full | Representations for the generalized inverses of a modified operator |
| title_fullStr | Representations for the generalized inverses of a modified operator |
| title_full_unstemmed | Representations for the generalized inverses of a modified operator |
| title_short | Representations for the generalized inverses of a modified operator |
| title_sort | representations for the generalized inverses of a modified operator |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1885 |
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