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Perturbation of Generalized Inverses of Linear Operators in Hilbert Spaces
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
198, 506]515 Ž1996.
0095
Perturbation of Generalized Inverses of Linear Operators in Hilbert Spaces J. Ding* Department of Mathematics, The Uni¨ ersity of Southern Mississippi, Hattiesburg, Mississippi 39406-5045
and L. J. Huang Department of Mathematics, Rockford College, Rockford, Illinois 61108 Submitted by William F. Ames Received May 9, 1994
Let X and Y be Hilbert spaces, and let T : X ª Y be a bounded linear operator with closed range. In this paper we present some new results of the perturbation analysis on the generalized inverse of T and the related least squares solution to the operator equation Tx s y. Q 1996 Academic Press, Inc.
1. INTRODUCTION Let X and Y be two Hilbert spaces, and let T : X ª Y be a bounded linear operator with closed range. In this paper, we shall consider the perturbation problem for the generalized inverse of T and the related problem min 5 x 5 , subject to 5 Tx y y 5 s min 5 Tz y y 5 . Ž 1. zgX
Nashed w7x has given a systematical investigation to the perturbation and approximations of generalized inverses of linear operators between Banach spaces. In w2x some perturbation analysis was presented for the least squares problem for bounded linear operators of special types between Hilbert spaces. In this paper we shall explore the more general perturbation problem. In the next section we introduce some concepts for * This paper is dedicated to the memory of this author’s teacher, Professor Xuchu He, who passed away on April 30, 1990. 506 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
INVERSES OF LINEAR OPERATORS
507
the later use. Section 3 is devoted to the perturbation analysis of the generalized inverse. Section 4 presents the corresponding error analysis for the least squares solution to Ž1.. We conclude in Section 5.
2. PRELIMINARIES Throughout this paper we assume that X and Y are complex Hilbert spaces. Let B Ž X, Y . be the Banach space of all bounded linear operators T : X ª Y with the operator norm 5 T 5 s sup5 Tx 5 : 5 x 5 s 14 , where 5 5 is the norm of X or Y induced by their respective inner product Ž?, ? .. We let RŽT . and N ŽT . denote the range and the null space of T, respectively. If A is a subspace of X, then AH is the orthogonal complement of A in X. Let T g B Ž X, Y . with closed range. The following definition is basically due to Y.-Y. Tseng Žsee w5x.. DEFINITION 2.1. The bounded linear operator T †: Y ª X defined by T † Tx s x
for x g N Ž T .
T †y s 0
for y g R Ž T .
H
and H
is called the generalized inverse of T. It is well known that x s T † y is the unique solution of the problem Ž1.. The proof of the following lemma can be found in w5x. LEMMA 2.1. Ži .
Let T g B Ž X, Y . with closed range. Then,
T † is the unique operator in B Ž Y, X . satisfying T † T s PRŽT † . ,
TT † s PRŽT . ,
where PM is the orthogonal projector on M. Žii. T † is the unique operator in B Ž Y, X . satisfying T † T s Ž T † T . *,
TT † s Ž TT † . *,
T † TT † s T † ,
TT † T s T ,
where T * is the adjoint of T. Žiii. ŽT † . † s T and ŽT *. † s ŽT † .*. The definition of the following number will play a key role in the paper.
508
DING AND HUANG
DEFINITION 2.2. Let T : X ª Y be a linear operator. Define
g Ž T . s inf 5 Tx 5 : x g N Ž T .
H
, 5 x 5 s 14 .
We list some useful properties of g in the following. For more details, see the monograph of Kato w6x. LEMMA 2.2.
Let T g B Ž X, Y .. Then
Ži. g ŽT . ) 0 if and only if RŽT . is closed. Žii. g ŽT *. s g ŽT .. Žiii. 5 T † 5 s g ŽT .y1 if RŽT . is closed. Proof. The proof of Ži. and Žii. can be found in w6x, and w2x is referred to for the proof of Žiii.. Q.E.D. 3. THE PERTURBATION ANALYSIS OF T † In this section we consider the perturbation problem of the generalized inverse of T g B Ž X, Y .. For this purpose, we need the concept of distance between two closed subspaces in a Banach space w6x. Suppose X is a Banach space. For x g X and M : X, let distŽ x, M . s inf y g M 5 x y y 5 and SM s x g M : 5 x 5 s 14 , and let M be the closure of M in X. Let A and B be two subspaces of X. Denote d Ž A, B . s sup x g S A distŽ x, B .. Similarly denote dŽ A, B . s sup x g S A distŽ x, SB .. Some properties of d Ž A, B . and dŽ A, B . are listed as follows; the proof is referred to w6x. LEMMA 3.1. Let X be a Banach space, and let A and B be closed subspaces of X. Then Ži. 0 F d Ž A, B . F d Ž A, B . F 2 d Ž A, B . F 2. Thus, lim n ª ` Ž d A n , Bn . s 0 if and only if lim n ª` dŽ A n , Bn . s 0. Žii. d Ž A, B . s d Ž B H , AH .. Žiii. d Ž A, B . - 1 implies dim A F dim B. LEMMA 3.2.
Let A and B be subspaces of a Banach space X. Then
d Ž A, B . s d Ž A, B . ,
d Ž A, B . s d Ž A, B . .
Proof. We only prove the first equality; the second one can be shown exactly the same way. For any given e ) 0, there exists an x g SA for
509
INVERSES OF LINEAR OPERATORS
which d Ž A, B . - distŽ x, B . q e . By the continuity of distance function, distŽ x, B . s distŽ x, B .. Hence
d Ž A, B . - dist Ž x, B . q e s dist Ž x, B . q e F d Ž A, B . q e . Since e is arbitrary, d Ž A, B . F d Ž A, B .. To establish the opposite inequality, let e ) 0 be fixed. There exists x g SA for which
d Ž A, B . - dist Ž x, B . q er2. Then there exists an x g SA such that 5 x y x 5 - er2. Therefore,
d Ž A, B . - dist Ž x, B . q
e 2
F 5 x y x 5 q dist Ž x, B . q
e 2
- dist Ž x, B . q e F d Ž A, B . q e F d Ž A, B . q e gives the required result.
Q.E.D.
Remark 3.1. Lemma 3.2 indicates that Ži. and Žii. in Lemma 3.1 are still valid without the closedness assumption for A and B. Moreover, Žiii. is also true in this case, based on Lemma IV.2.3 of w6x. Now let X and Y be Hilbert spaces, and let T, T s T q d T g B Ž X, Y .. LEMMA 3.3.
g Ž T . G g Ž T . y 5T 5 d N Ž T .
ž
H
, NŽT .
H
/ y 5dT 5
G g Ž T . y 2 5T 5 d Ž N Ž T . , N Ž T . . y 5 d T 5;
g Ž T . G g Ž T . y 5T 5 dŽ RŽ T . , RŽ T . . y 5 d T 5 G g Ž T . y 2 5T 5 d Ž RŽ T . , RŽ T . . y 5 d T 5. Proof. From the definition of g , we have
g Ž T . s inf 5 Tx 5 : x g SNŽT . H s inf 5 Tx q d Tx 5 : x g SNŽT . H 4
½
5
G inf 5 Tx 5 y 5 d Tx 5 : x g SNŽT . H 4 G inf 5 Tx 5 : x g SNŽT . H 4 y 5 d T 5 .
Ž 2. Ž 3. Ž 4. Ž 5.
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DING AND HUANG
Now fix x g S NŽT . H . Given any e ) 0, there is y g SNŽT . H such that 5 x y y 5 F distŽ x, S NŽT . H . q e . Hence, 5 Tx 5 s 5 Ty q Tx y Ty 5 G 5 Ty 5 y 5 Tx y Ty 5 G g Ž T . y 5 T 5 5 x y y 5 G g Ž T . y 5 T 5 dist Ž x, SNŽT . H . q e G g Ž T . y 5T 5 d N Ž T .
ž
H
H
, NŽT .
/ qe
.
Since e is arbitrary, we have 5 Tx 5 G g Ž T . y 5 T 5 d N Ž T .
ž
H
, NŽT .
H
/.
Noting that x g SNŽT . H is arbitrary, and using Lemma 3.1, we have
g Ž T . G g Ž T . y 5T 5 d N Ž T .
ž
H
G g Ž T . y 2 5T 5 d N Ž T .
ž
H
H
/ y 5dT 5 , NŽT . / y 5d T 5
, NŽT .
H
s g Ž T . y 2 5T 5 d Ž N Ž T . , N Ž T . . y 5 d T 5. This proves Ž2. and Ž3.. To prove Ž4. and Ž5., noting N ŽT *. H s R Ž T . and N ŽT *. H s R Ž T . , applying Ž2. and Ž3. to T * and T *, and using Lemmas 2.2 and 3.2, we obtain
g Ž T . s g Ž T *. G g Ž T *. y 5 T *5 d N Ž T *.
ž
H
, N Ž T *.
H
/ y 5 d T *5
s g Ž T . y 5T 5 d ŽRŽ T . , RŽ T . . y 5 d T 5 s g Ž T . y 5T 5 dŽ RŽ T . , RŽ T . . y 5 d T 5 G g Ž T . y 2 5T 5 d Ž RŽ T . , RŽ T . . y 5 d T 5.
Q.E.D.
Using the gap function d directly, we have another lower bound for g ŽT .. LEMMA 3.4.
If d Ž N ŽT ., N ŽT .. - 1, then
g ŽT . G g ŽT . 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
y 5 d T 5.
Ž 6.
y 5 d T 5.
Ž 7.
If d Ž RŽT ., RŽT .. - 1, then
g Ž T . G g Ž T . 1 y d Ž RŽ T . , RŽ T . .
½
2 1r2
5
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INVERSES OF LINEAR OPERATORS
Proof. From the proof of Lemma 3.3,
g Ž T . G inf 5 Tx 5 : x g SNŽT . H 4 y 5 d T 5 . Now fix x g S NŽT . H . Then there is a unique y g N ŽT . H such that 5 x y y 5 s dist Ž x, N Ž T . H . . It follows that x y y g N ŽT .. Hence 5 Tx 5 s 5 Ty 5. From 5 y 5 2 s 5 x 5 2 y 5 x y y 5 2 s 1 y wdistŽ x, N ŽT . H.x 2 , we have 5 y5 s 1 y d NŽT .
½
ž
H
, NŽT .
H
2 1r2
/
5
s 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
.
Now d Ž N ŽT ., N ŽT .. - 1 implies 5 y 5 ) 0. Hence, 5 Tx 5 s
5 Ty 5 5 y5
5 y5 s
5 Ty 5
1y 5 y5 ½
d Ž NŽT . , NŽT . .
G g ŽT . 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
2 1r2
5
.
Since x g SNŽT . H is arbitrary, we have
g ŽT . G g ŽT . 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
y 5 d T 5,
which is Ž6., and Ž7. is from applying Ž6. to T *.
Q.E.D.
Remark 3.2. By Žiii. of Lemma 3.1 and Remark 3.1, the condition d Ž N ŽT ., N ŽT .. - 1 or d Ž RŽT ., RŽT .. - 1 in the above lemma implies that dim N ŽT . F dim N ŽT . and dim RŽT . F dim RŽT ., respectively. Now it is easy to prove the following perturbation theorem for T †. In the remainder of this paper, we let d N s d Ž N ŽT ., N ŽT .. and d R s d Ž RŽT ., RŽT ... Let k s k ŽT . s 5 T 5 5 T † 5. THEOREM 3.1. Let X and Y be Hilbert spaces, let T g B Ž X, Y . with closed range, and let T s T q d T g B Ž X, Y .. Then Ži. 2 kd N q 5 d T 5 5 T † 5 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5 1 y 2 kd N y 5 d T 5 5 T † 5
,
Ž 8.
and 2 kd R q 5 d T 5 5 T † 5 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5 1 y 2 kd R y 5 d T 5 5 T † 5
.
Ž 9.
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DING AND HUANG
Žii.
d N2 q 5 d T 5 2 5 T † 5 2 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5
Ž 1 y dN2 .
1r2
y 5 d T 5 5T † 5
,
Ž 10 .
.
Ž 11 .
and d R2 q 5 d T 5 2 5 T † 5 2 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5
Ž 1 y dR2 .
1r2
y 5 d T 5 5T † 5
Proof. The condition in Ži. or Žii. implies g ŽT . ) 0. Hence RŽT . is closed, and T † is well defined. Noting that 5 T † 5 s g ŽT .y1 and 5 T † 5 s g ŽT .y1 , and using Lemma 3.3 and Lemma 3.4 for Ži. and Žii., respectively, we obtain the theorem. Q.E.D. Remark 3.3. Since k G 1, the condition in Theorem 3.1 implies d N - 1 or d R - 1, and thus dim N ŽT . F dim N ŽT . or dim RŽT . F dim RŽT ., respectively. Actually dim N ŽT . s dim N ŽT . or dim RŽT . s dim RŽT . since 5 d T 5 5 T † 5 - 1 implies dim N ŽT . F dim N ŽT . by Theorem V.1.2 of w3x.
4. ERROR ESTIMATES FOR LEAST SQUARES PROBLEMS Using the results of the previous section, we can present the error estimates for the least squares solution of the problem Ž1.. We need the following decomposition result whose proof is referred to w7x. LEMMA 4.1.
Let T and T s T q d T g B Ž X, Y . with closed ranges. Then
T † y T † s yT †d TT † q T † Ž T † . * Ž d T . * Ž I y TT † . q Ž I y T † T . Ž d T . * Ž T † . *T † . Remark 4.1. becomes
Ž 12 .
N ŽT . : N ŽT . if and only if Ž I y T † T .Ž d T .* s 0, and Ž12.
T † y T † s yT †d TT † q T † Ž T † . * Ž d T . * Ž I y TT † . . RŽT . : RŽT . if and only if Ž d T .*Ž I y TT † . s 0, and Ž12. is reduced to T † y T † s yT †d TT † q Ž I y T † T . Ž d T . * Ž T † . *T † . Now let T g B Ž X, Y . with closed range, let T s T q d T g B Ž X, Y ., and let y, y s y q d y g Y. Denote e T s 5 d T 5r5 T 5 and e y s 5 d y 5r5 y 5. In the following we let x s T † y / 0 and x s T † y.
513
INVERSES OF LINEAR OPERATORS
THEOREM 4.1. 5 x y x5 5 x5
F
If k Ž2 d N q e T . - 1, then
k 1 y k Ž 2 dN q e T . q
½
ey
5 y5
q eT
5T 5 5 x 5
5 y y Tx 5
ke T
1 y k Ž 2 dN q e T . 5 T 5 5 x 5
5
q ke T . Ž 13 .
5
q ke T . Ž 14 .
If k Ž2 d R q e T . - 1, then 5 x y x5 5 x5
F
k 1 y k Ž 2 dR q e T . q
½
ey
5 y5 5T 5 5 x 5
q eT 5 y y Tx 5
ke T
1 y k Ž 2 dR q e T . 5 T 5 5 x 5
Proof. It is enough to prove Ž13.. From Lemma 4.1 and Ži. of Theorem 3.1, 5 x y x 5 s 5 T † y y T † y 5 s 5 T †d y q Ž T † y T † . y 5 s 5 T †d y y T †d TT † y q T † Ž T † . * Ž d T . * Ž I y TT † . y q Ž I y T † T . Ž d T . * Ž T † . *T † y 5 F 5T † 5 5 d y 5 q 5T † 5 5 d T 5 5 x 5 q 5 T † 5 2 5 d T 5 5 y y Tx 5 q 5 d T 5 5 T † 5 5 x 5 F
5T † 5 1 y 2 kd N y 5 d T 5 5 T † 5 q q
s
½
ey 5 y5
5T † 5 1 y 2 kd N y 5 d T 5 5 T † 5
eT 5T 5 5 x 5
5T † 5 2
Ž 1 y2 kdN y 5 d T 55 T † 5 . k
1 y k Ž 2 dN q e T . q
This completes the proof.
ey
2
e T 5 T 55 y y Tx 5 q e T 5 T 55 T † 55 x 5
5 y5 5T 5 5 x 5
q eT
ke T
5 y y Tx 5
1 y k Ž 2 dN q e T . 5 T 5 5 x 5
5
q ke T 5 x 5 . Q.E.D.
514
DING AND HUANG
If in addition y g RŽT . in the abo¨ e theorem, then
COROLLARY 4.1.
5 x y x5 5 x5
F
k 1 y k Ž 2 d q eT .
e y q e T 4 q ke T ,
Ž 15 .
where d s d N if k Ž2 d N q e T . - 1 and d s d R if k Ž2 d R q e T . - 1. Similarly, using Žii. of Theorem 3.1, we can prove the following. If d N2 q Ž ke T . 2 - 1, then
THEOREM 4.2. 5 x y x5 5 x5
F
k
Ž1 y
1r2 d N2
y ke T
.
q
½
ey
5 y5 5T 5 5 x 5
q eT 5 y y Tx 5
ke T
Ž1 y
1r2 d N2
.
y ke T 5 T 5 5 x 5
5
q ke T . Ž 16 .
5
q ke T . Ž 17 .
If d R2 q Ž ke T . 2 - 1, then 5 x y x5 5 x5
F
k 1r2 d R2
Ž1 y .
y ke T q
COROLLARY 4.2. 5 x y x5 5 x5
½
ey
5 y5 5T 5 5 x 5
q eT 5 y y Tx 5
ke T 1r2 d R2
Ž1 y .
y ke T 5 T 5 5 x 5
If in addition y g RŽT . in the abo¨ e theorem, then F
k
Ž1 y d 2 .
1r2
y ke T
e y q e T 4 q ke T ,
Ž 18 .
where d s d N if d N2 q Ž ke T . 2 - 1 and d s d R if d R2 q Ž ke T . 2 - 1. 5. CONCLUSIONS In this paper we extended some of the results in the perturbation analysis of matrices to linear operators on Hilbert spaces, using the general perturbation theory for linear operator as developed in w6x. Error estimates have been obtained for the perturbation which keeps the dimension of the null space or the range of the operator. A simple consequence is that the generalized inverse is continuous if the perturbation is either not dimension-increasing with respect to the null space of the operator or
INVERSES OF LINEAR OPERATORS
515
not dimension-decreasing with respect to the range of the operator. For the arbitrary perturbation, however, what we can expect is only the upper semi-continuity as indicated in w1x for matrices. Compared with the perturbation results for matrices, the right side of the estimates here for infinitely dimensional linear operators contains the distance d N or d R as well as 5 d T 5. It should be interesting to explore the relation among such quantities.
REFERENCES 1. J. Ding, The upper semicontinuity of the least squares problems, in ‘‘Progress in Systems and Control Theory V,’’ pp. 458]467, Birkhauser, Boston, 1990. ¨ 2. J. Ding and L. J. Huang, On the perturbation of the least squares solutions in Hilbert spaces, Linear Algebra Appl. 212 r 213 Ž1994., 487]500. 3. S. Goldberg, ‘‘Unbounded Linear Operators, Theory and Applications,’’ McGraw]Hill, New York, 1966. 4. C. W. Groetsch, ‘‘Generalized Inverses of Linear Operators,’’ Dekker, New York, 1977. 5. A. Ben-Israel and T. N. E. Greville, ‘‘Generalized Inverses: Theory and Applications,’’ Wiley, New York, 1974. 6. T. Kato, ‘‘Perturbation Theory for Linear Operators,’’ Springer-Verlag, New York, 1984. 7. M. Z. Nashed, Perturbations and approximations for generalized inverses and linear operator equations, in ‘‘Generalized Inverses and Applications’’ ŽM. Z. Nashed, Ed.., pp. 325]396, Academic Press, New YorkrLondon, 1976. 8. G. W. Stewart and J.-G. Sun, ‘‘Matrix Perturbation Theory,’’ Academic Press, New YorkrLondon, 1990. ˚ Wedin, Perturbation theory for pseudo-inverses, BIT 13 Ž1973., 217]232. 9. P.-A.
198, 506]515 Ž1996.
0095
Perturbation of Generalized Inverses of Linear Operators in Hilbert Spaces J. Ding* Department of Mathematics, The Uni¨ ersity of Southern Mississippi, Hattiesburg, Mississippi 39406-5045
and L. J. Huang Department of Mathematics, Rockford College, Rockford, Illinois 61108 Submitted by William F. Ames Received May 9, 1994
Let X and Y be Hilbert spaces, and let T : X ª Y be a bounded linear operator with closed range. In this paper we present some new results of the perturbation analysis on the generalized inverse of T and the related least squares solution to the operator equation Tx s y. Q 1996 Academic Press, Inc.
1. INTRODUCTION Let X and Y be two Hilbert spaces, and let T : X ª Y be a bounded linear operator with closed range. In this paper, we shall consider the perturbation problem for the generalized inverse of T and the related problem min 5 x 5 , subject to 5 Tx y y 5 s min 5 Tz y y 5 . Ž 1. zgX
Nashed w7x has given a systematical investigation to the perturbation and approximations of generalized inverses of linear operators between Banach spaces. In w2x some perturbation analysis was presented for the least squares problem for bounded linear operators of special types between Hilbert spaces. In this paper we shall explore the more general perturbation problem. In the next section we introduce some concepts for * This paper is dedicated to the memory of this author’s teacher, Professor Xuchu He, who passed away on April 30, 1990. 506 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
INVERSES OF LINEAR OPERATORS
507
the later use. Section 3 is devoted to the perturbation analysis of the generalized inverse. Section 4 presents the corresponding error analysis for the least squares solution to Ž1.. We conclude in Section 5.
2. PRELIMINARIES Throughout this paper we assume that X and Y are complex Hilbert spaces. Let B Ž X, Y . be the Banach space of all bounded linear operators T : X ª Y with the operator norm 5 T 5 s sup5 Tx 5 : 5 x 5 s 14 , where 5 5 is the norm of X or Y induced by their respective inner product Ž?, ? .. We let RŽT . and N ŽT . denote the range and the null space of T, respectively. If A is a subspace of X, then AH is the orthogonal complement of A in X. Let T g B Ž X, Y . with closed range. The following definition is basically due to Y.-Y. Tseng Žsee w5x.. DEFINITION 2.1. The bounded linear operator T †: Y ª X defined by T † Tx s x
for x g N Ž T .
T †y s 0
for y g R Ž T .
H
and H
is called the generalized inverse of T. It is well known that x s T † y is the unique solution of the problem Ž1.. The proof of the following lemma can be found in w5x. LEMMA 2.1. Ži .
Let T g B Ž X, Y . with closed range. Then,
T † is the unique operator in B Ž Y, X . satisfying T † T s PRŽT † . ,
TT † s PRŽT . ,
where PM is the orthogonal projector on M. Žii. T † is the unique operator in B Ž Y, X . satisfying T † T s Ž T † T . *,
TT † s Ž TT † . *,
T † TT † s T † ,
TT † T s T ,
where T * is the adjoint of T. Žiii. ŽT † . † s T and ŽT *. † s ŽT † .*. The definition of the following number will play a key role in the paper.
508
DING AND HUANG
DEFINITION 2.2. Let T : X ª Y be a linear operator. Define
g Ž T . s inf 5 Tx 5 : x g N Ž T .
H
, 5 x 5 s 14 .
We list some useful properties of g in the following. For more details, see the monograph of Kato w6x. LEMMA 2.2.
Let T g B Ž X, Y .. Then
Ži. g ŽT . ) 0 if and only if RŽT . is closed. Žii. g ŽT *. s g ŽT .. Žiii. 5 T † 5 s g ŽT .y1 if RŽT . is closed. Proof. The proof of Ži. and Žii. can be found in w6x, and w2x is referred to for the proof of Žiii.. Q.E.D. 3. THE PERTURBATION ANALYSIS OF T † In this section we consider the perturbation problem of the generalized inverse of T g B Ž X, Y .. For this purpose, we need the concept of distance between two closed subspaces in a Banach space w6x. Suppose X is a Banach space. For x g X and M : X, let distŽ x, M . s inf y g M 5 x y y 5 and SM s x g M : 5 x 5 s 14 , and let M be the closure of M in X. Let A and B be two subspaces of X. Denote d Ž A, B . s sup x g S A distŽ x, B .. Similarly denote dŽ A, B . s sup x g S A distŽ x, SB .. Some properties of d Ž A, B . and dŽ A, B . are listed as follows; the proof is referred to w6x. LEMMA 3.1. Let X be a Banach space, and let A and B be closed subspaces of X. Then Ži. 0 F d Ž A, B . F d Ž A, B . F 2 d Ž A, B . F 2. Thus, lim n ª ` Ž d A n , Bn . s 0 if and only if lim n ª` dŽ A n , Bn . s 0. Žii. d Ž A, B . s d Ž B H , AH .. Žiii. d Ž A, B . - 1 implies dim A F dim B. LEMMA 3.2.
Let A and B be subspaces of a Banach space X. Then
d Ž A, B . s d Ž A, B . ,
d Ž A, B . s d Ž A, B . .
Proof. We only prove the first equality; the second one can be shown exactly the same way. For any given e ) 0, there exists an x g SA for
509
INVERSES OF LINEAR OPERATORS
which d Ž A, B . - distŽ x, B . q e . By the continuity of distance function, distŽ x, B . s distŽ x, B .. Hence
d Ž A, B . - dist Ž x, B . q e s dist Ž x, B . q e F d Ž A, B . q e . Since e is arbitrary, d Ž A, B . F d Ž A, B .. To establish the opposite inequality, let e ) 0 be fixed. There exists x g SA for which
d Ž A, B . - dist Ž x, B . q er2. Then there exists an x g SA such that 5 x y x 5 - er2. Therefore,
d Ž A, B . - dist Ž x, B . q
e 2
F 5 x y x 5 q dist Ž x, B . q
e 2
- dist Ž x, B . q e F d Ž A, B . q e F d Ž A, B . q e gives the required result.
Q.E.D.
Remark 3.1. Lemma 3.2 indicates that Ži. and Žii. in Lemma 3.1 are still valid without the closedness assumption for A and B. Moreover, Žiii. is also true in this case, based on Lemma IV.2.3 of w6x. Now let X and Y be Hilbert spaces, and let T, T s T q d T g B Ž X, Y .. LEMMA 3.3.
g Ž T . G g Ž T . y 5T 5 d N Ž T .
ž
H
, NŽT .
H
/ y 5dT 5
G g Ž T . y 2 5T 5 d Ž N Ž T . , N Ž T . . y 5 d T 5;
g Ž T . G g Ž T . y 5T 5 dŽ RŽ T . , RŽ T . . y 5 d T 5 G g Ž T . y 2 5T 5 d Ž RŽ T . , RŽ T . . y 5 d T 5. Proof. From the definition of g , we have
g Ž T . s inf 5 Tx 5 : x g SNŽT . H s inf 5 Tx q d Tx 5 : x g SNŽT . H 4
½
5
G inf 5 Tx 5 y 5 d Tx 5 : x g SNŽT . H 4 G inf 5 Tx 5 : x g SNŽT . H 4 y 5 d T 5 .
Ž 2. Ž 3. Ž 4. Ž 5.
510
DING AND HUANG
Now fix x g S NŽT . H . Given any e ) 0, there is y g SNŽT . H such that 5 x y y 5 F distŽ x, S NŽT . H . q e . Hence, 5 Tx 5 s 5 Ty q Tx y Ty 5 G 5 Ty 5 y 5 Tx y Ty 5 G g Ž T . y 5 T 5 5 x y y 5 G g Ž T . y 5 T 5 dist Ž x, SNŽT . H . q e G g Ž T . y 5T 5 d N Ž T .
ž
H
H
, NŽT .
/ qe
.
Since e is arbitrary, we have 5 Tx 5 G g Ž T . y 5 T 5 d N Ž T .
ž
H
, NŽT .
H
/.
Noting that x g SNŽT . H is arbitrary, and using Lemma 3.1, we have
g Ž T . G g Ž T . y 5T 5 d N Ž T .
ž
H
G g Ž T . y 2 5T 5 d N Ž T .
ž
H
H
/ y 5dT 5 , NŽT . / y 5d T 5
, NŽT .
H
s g Ž T . y 2 5T 5 d Ž N Ž T . , N Ž T . . y 5 d T 5. This proves Ž2. and Ž3.. To prove Ž4. and Ž5., noting N ŽT *. H s R Ž T . and N ŽT *. H s R Ž T . , applying Ž2. and Ž3. to T * and T *, and using Lemmas 2.2 and 3.2, we obtain
g Ž T . s g Ž T *. G g Ž T *. y 5 T *5 d N Ž T *.
ž
H
, N Ž T *.
H
/ y 5 d T *5
s g Ž T . y 5T 5 d ŽRŽ T . , RŽ T . . y 5 d T 5 s g Ž T . y 5T 5 dŽ RŽ T . , RŽ T . . y 5 d T 5 G g Ž T . y 2 5T 5 d Ž RŽ T . , RŽ T . . y 5 d T 5.
Q.E.D.
Using the gap function d directly, we have another lower bound for g ŽT .. LEMMA 3.4.
If d Ž N ŽT ., N ŽT .. - 1, then
g ŽT . G g ŽT . 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
y 5 d T 5.
Ž 6.
y 5 d T 5.
Ž 7.
If d Ž RŽT ., RŽT .. - 1, then
g Ž T . G g Ž T . 1 y d Ž RŽ T . , RŽ T . .
½
2 1r2
5
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INVERSES OF LINEAR OPERATORS
Proof. From the proof of Lemma 3.3,
g Ž T . G inf 5 Tx 5 : x g SNŽT . H 4 y 5 d T 5 . Now fix x g S NŽT . H . Then there is a unique y g N ŽT . H such that 5 x y y 5 s dist Ž x, N Ž T . H . . It follows that x y y g N ŽT .. Hence 5 Tx 5 s 5 Ty 5. From 5 y 5 2 s 5 x 5 2 y 5 x y y 5 2 s 1 y wdistŽ x, N ŽT . H.x 2 , we have 5 y5 s 1 y d NŽT .
½
ž
H
, NŽT .
H
2 1r2
/
5
s 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
.
Now d Ž N ŽT ., N ŽT .. - 1 implies 5 y 5 ) 0. Hence, 5 Tx 5 s
5 Ty 5 5 y5
5 y5 s
5 Ty 5
1y 5 y5 ½
d Ž NŽT . , NŽT . .
G g ŽT . 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
2 1r2
5
.
Since x g SNŽT . H is arbitrary, we have
g ŽT . G g ŽT . 1 y d Ž NŽT . , NŽT . .
½
2 1r2
5
y 5 d T 5,
which is Ž6., and Ž7. is from applying Ž6. to T *.
Q.E.D.
Remark 3.2. By Žiii. of Lemma 3.1 and Remark 3.1, the condition d Ž N ŽT ., N ŽT .. - 1 or d Ž RŽT ., RŽT .. - 1 in the above lemma implies that dim N ŽT . F dim N ŽT . and dim RŽT . F dim RŽT ., respectively. Now it is easy to prove the following perturbation theorem for T †. In the remainder of this paper, we let d N s d Ž N ŽT ., N ŽT .. and d R s d Ž RŽT ., RŽT ... Let k s k ŽT . s 5 T 5 5 T † 5. THEOREM 3.1. Let X and Y be Hilbert spaces, let T g B Ž X, Y . with closed range, and let T s T q d T g B Ž X, Y .. Then Ži. 2 kd N q 5 d T 5 5 T † 5 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5 1 y 2 kd N y 5 d T 5 5 T † 5
,
Ž 8.
and 2 kd R q 5 d T 5 5 T † 5 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5 1 y 2 kd R y 5 d T 5 5 T † 5
.
Ž 9.
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DING AND HUANG
Žii.
d N2 q 5 d T 5 2 5 T † 5 2 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5
Ž 1 y dN2 .
1r2
y 5 d T 5 5T † 5
,
Ž 10 .
.
Ž 11 .
and d R2 q 5 d T 5 2 5 T † 5 2 - 1 implies RŽT . is closed and 5T † 5 F
5T † 5
Ž 1 y dR2 .
1r2
y 5 d T 5 5T † 5
Proof. The condition in Ži. or Žii. implies g ŽT . ) 0. Hence RŽT . is closed, and T † is well defined. Noting that 5 T † 5 s g ŽT .y1 and 5 T † 5 s g ŽT .y1 , and using Lemma 3.3 and Lemma 3.4 for Ži. and Žii., respectively, we obtain the theorem. Q.E.D. Remark 3.3. Since k G 1, the condition in Theorem 3.1 implies d N - 1 or d R - 1, and thus dim N ŽT . F dim N ŽT . or dim RŽT . F dim RŽT ., respectively. Actually dim N ŽT . s dim N ŽT . or dim RŽT . s dim RŽT . since 5 d T 5 5 T † 5 - 1 implies dim N ŽT . F dim N ŽT . by Theorem V.1.2 of w3x.
4. ERROR ESTIMATES FOR LEAST SQUARES PROBLEMS Using the results of the previous section, we can present the error estimates for the least squares solution of the problem Ž1.. We need the following decomposition result whose proof is referred to w7x. LEMMA 4.1.
Let T and T s T q d T g B Ž X, Y . with closed ranges. Then
T † y T † s yT †d TT † q T † Ž T † . * Ž d T . * Ž I y TT † . q Ž I y T † T . Ž d T . * Ž T † . *T † . Remark 4.1. becomes
Ž 12 .
N ŽT . : N ŽT . if and only if Ž I y T † T .Ž d T .* s 0, and Ž12.
T † y T † s yT †d TT † q T † Ž T † . * Ž d T . * Ž I y TT † . . RŽT . : RŽT . if and only if Ž d T .*Ž I y TT † . s 0, and Ž12. is reduced to T † y T † s yT †d TT † q Ž I y T † T . Ž d T . * Ž T † . *T † . Now let T g B Ž X, Y . with closed range, let T s T q d T g B Ž X, Y ., and let y, y s y q d y g Y. Denote e T s 5 d T 5r5 T 5 and e y s 5 d y 5r5 y 5. In the following we let x s T † y / 0 and x s T † y.
513
INVERSES OF LINEAR OPERATORS
THEOREM 4.1. 5 x y x5 5 x5
F
If k Ž2 d N q e T . - 1, then
k 1 y k Ž 2 dN q e T . q
½
ey
5 y5
q eT
5T 5 5 x 5
5 y y Tx 5
ke T
1 y k Ž 2 dN q e T . 5 T 5 5 x 5
5
q ke T . Ž 13 .
5
q ke T . Ž 14 .
If k Ž2 d R q e T . - 1, then 5 x y x5 5 x5
F
k 1 y k Ž 2 dR q e T . q
½
ey
5 y5 5T 5 5 x 5
q eT 5 y y Tx 5
ke T
1 y k Ž 2 dR q e T . 5 T 5 5 x 5
Proof. It is enough to prove Ž13.. From Lemma 4.1 and Ži. of Theorem 3.1, 5 x y x 5 s 5 T † y y T † y 5 s 5 T †d y q Ž T † y T † . y 5 s 5 T †d y y T †d TT † y q T † Ž T † . * Ž d T . * Ž I y TT † . y q Ž I y T † T . Ž d T . * Ž T † . *T † y 5 F 5T † 5 5 d y 5 q 5T † 5 5 d T 5 5 x 5 q 5 T † 5 2 5 d T 5 5 y y Tx 5 q 5 d T 5 5 T † 5 5 x 5 F
5T † 5 1 y 2 kd N y 5 d T 5 5 T † 5 q q
s
½
ey 5 y5
5T † 5 1 y 2 kd N y 5 d T 5 5 T † 5
eT 5T 5 5 x 5
5T † 5 2
Ž 1 y2 kdN y 5 d T 55 T † 5 . k
1 y k Ž 2 dN q e T . q
This completes the proof.
ey
2
e T 5 T 55 y y Tx 5 q e T 5 T 55 T † 55 x 5
5 y5 5T 5 5 x 5
q eT
ke T
5 y y Tx 5
1 y k Ž 2 dN q e T . 5 T 5 5 x 5
5
q ke T 5 x 5 . Q.E.D.
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DING AND HUANG
If in addition y g RŽT . in the abo¨ e theorem, then
COROLLARY 4.1.
5 x y x5 5 x5
F
k 1 y k Ž 2 d q eT .
e y q e T 4 q ke T ,
Ž 15 .
where d s d N if k Ž2 d N q e T . - 1 and d s d R if k Ž2 d R q e T . - 1. Similarly, using Žii. of Theorem 3.1, we can prove the following. If d N2 q Ž ke T . 2 - 1, then
THEOREM 4.2. 5 x y x5 5 x5
F
k
Ž1 y
1r2 d N2
y ke T
.
q
½
ey
5 y5 5T 5 5 x 5
q eT 5 y y Tx 5
ke T
Ž1 y
1r2 d N2
.
y ke T 5 T 5 5 x 5
5
q ke T . Ž 16 .
5
q ke T . Ž 17 .
If d R2 q Ž ke T . 2 - 1, then 5 x y x5 5 x5
F
k 1r2 d R2
Ž1 y .
y ke T q
COROLLARY 4.2. 5 x y x5 5 x5
½
ey
5 y5 5T 5 5 x 5
q eT 5 y y Tx 5
ke T 1r2 d R2
Ž1 y .
y ke T 5 T 5 5 x 5
If in addition y g RŽT . in the abo¨ e theorem, then F
k
Ž1 y d 2 .
1r2
y ke T
e y q e T 4 q ke T ,
Ž 18 .
where d s d N if d N2 q Ž ke T . 2 - 1 and d s d R if d R2 q Ž ke T . 2 - 1. 5. CONCLUSIONS In this paper we extended some of the results in the perturbation analysis of matrices to linear operators on Hilbert spaces, using the general perturbation theory for linear operator as developed in w6x. Error estimates have been obtained for the perturbation which keeps the dimension of the null space or the range of the operator. A simple consequence is that the generalized inverse is continuous if the perturbation is either not dimension-increasing with respect to the null space of the operator or
INVERSES OF LINEAR OPERATORS
515
not dimension-decreasing with respect to the range of the operator. For the arbitrary perturbation, however, what we can expect is only the upper semi-continuity as indicated in w1x for matrices. Compared with the perturbation results for matrices, the right side of the estimates here for infinitely dimensional linear operators contains the distance d N or d R as well as 5 d T 5. It should be interesting to explore the relation among such quantities.
REFERENCES 1. J. Ding, The upper semicontinuity of the least squares problems, in ‘‘Progress in Systems and Control Theory V,’’ pp. 458]467, Birkhauser, Boston, 1990. ¨ 2. J. Ding and L. J. Huang, On the perturbation of the least squares solutions in Hilbert spaces, Linear Algebra Appl. 212 r 213 Ž1994., 487]500. 3. S. Goldberg, ‘‘Unbounded Linear Operators, Theory and Applications,’’ McGraw]Hill, New York, 1966. 4. C. W. Groetsch, ‘‘Generalized Inverses of Linear Operators,’’ Dekker, New York, 1977. 5. A. Ben-Israel and T. N. E. Greville, ‘‘Generalized Inverses: Theory and Applications,’’ Wiley, New York, 1974. 6. T. Kato, ‘‘Perturbation Theory for Linear Operators,’’ Springer-Verlag, New York, 1984. 7. M. Z. Nashed, Perturbations and approximations for generalized inverses and linear operator equations, in ‘‘Generalized Inverses and Applications’’ ŽM. Z. Nashed, Ed.., pp. 325]396, Academic Press, New YorkrLondon, 1976. 8. G. W. Stewart and J.-G. Sun, ‘‘Matrix Perturbation Theory,’’ Academic Press, New YorkrLondon, 1990. ˚ Wedin, Perturbation theory for pseudo-inverses, BIT 13 Ž1973., 217]232. 9. P.-A.