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Some equivalent conditions of stable perturbation of operators in Hilbert spaces
Applied Mathematics and Computation 147 (2004) 765–772 www.elsevier.com/locate/amc
Some equivalent conditions of stable perturbation of operators in Hilbert spaces q Yifeng Xue
a,*
, Guoliang Chen
b
a
b
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China Department of Mathematics, East China Normal University, Shanghai 200062, PR China
Abstract Let H1 ; H2 be two Hilbert spaces over the complex field C and let T ; T ¼ T þ dT : H1 ! H2 be two bounded linear operators with the M–P generalized inverse T þ . If RðT Þ \ RðT Þ? ¼ 0, we say that T is the stable perturbation of T . In this paper, we give five equivalent conditions that make T being the stable perturbation of T under the assumption kT þ kkdT k < 1. These equivalent conditions generalize not only the notation of rank-preserving perturbations of matrices but also the notation of acute perturbations of matrices. As a result, we obtain the following: Suppose that RðT Þ \ RðT Þ? ¼ 0 and kT þ kkdT k < 1. Then pffiffiffi 1þ 5 þ þ kT kkT þ kkdT k: kT T þ k 6 2 This result generalizes corresponding result when H1 ; H2 are all finite-dimensional. Ó 2003 Elsevier Inc. All rights reserved. Keywords: M–P generalized inverses; Stable perturbation of operators
q
Supported by the Natural Sciences Foundation, PR China. Corresponding author. E-mail address: [email protected] (Y. Xue).
*
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00810-X
766
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
1. Introduction Let ðH1 ; k kÞ; ðH2 ; k kÞ be two Hilbert spaces over the complex field C and let BðH1 ; H2 Þ denote the Banach space of all bounded linear operators T : H1 ! H2 with the norm kT k ¼ supfkTxk j kxk ¼ 1; x 2 H1 g: For T 2 BðH1 ; H2 Þ, Ker T (resp. RðT Þ) denotes the null space (resp. range) of T . According to [7], T 2 BðH1 ; H2 Þ with RðT Þ closed has a Moore–Penrose inverse T þ , namely, T þ is the unique solution for the equations T þ TT þ ¼ T þ ;
TT þ T ¼ T ;
T þT ¼ I P ;
TT þ ¼ Q;
ð1:1Þ
where P (resp. Q) is the orthogonal projection of H1 onto KerðT Þ (resp. H2 onto þ RðT Þ). We have ðT Þ ¼ ðT þ Þ , here T is the adjoint of T . In [1], Chen et al. proposed two notations about the perturbation of operators on Hilbert spaces, that is, for T ; T ¼ T þ dT 2 BðH1 ; H2 Þ, if RðT Þ \ ? ? RðT Þ ¼ 0 (resp. Ker T \ ðKer T Þ ¼ 0), then T is said to be the type I (resp. Type II) perturbation of T . They are the same when RðT Þ is closed and kT þ kkdT k < 12 (cf. [1, Corollary 3.2]). Later, Chen and Xue introduced a notation so-called the stable perturbation of operators on Banach spaces in [3], that is, if RðT Þ \ Ker T þ ¼ 0, then T is said to be the stable perturbation of T . But ? noting that Ker T þ ¼ ðI TT þ ÞH1 ¼ RðT Þ , we get that T is the stable perturbation of T iff T is the Type I perturbation of T . In this paper, authors will continue to investigate the conditions that make the perturbation of operators stable in Hilbert spaces. As a result, we get ? that if T ; T ¼ T þ dT 2 BðH1 ; H2 Þ with RðT Þ closed, RðT Þ \ RðT Þ ¼ 0 and þ þ kT kkdT k < 1, then T exists and pffiffiffi þ 1 þ kT T þ k 6 1 þ 5 kT kkT þ kkdT k: 2 This result generalized corresponding result in perturbation analysis of matrices with rank-preserving (cf. [9]). Finally, we consider the perturbation problem of the orthogonal projection of a point onto a linear manifold: min kp xk
subject to kTx bk ¼ min kTz bk; z2H1
ð1:2Þ
where p; x 2 H1 , b 2 H2 and T 2 BðH1 ; H2 Þ with RðT Þ closed. The perturbation analysis of Eq. (1.2) was discussed by Ding in [5] when H1 ; H2 are finitedimensional. When H1 ; H2 are all infinite-dimensional, the perturbation analysis of Eq. (1.2) was considered by Chen and Wei in [2], but their result is not good. So we will give a better estimate of the perturbation analysis of Eq. (1.2) than that of [2].
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
767
2. The main result Let M1 ; M2 be two closed subspaces in the Hilbert space H (with the norm k k). Set dðM1 ; M2 Þ ¼ supfdistðx; M2 Þ j kxk ¼ 1; x 2 M1 g: If M1 ¼ 0, we set dðM1 ; M2 Þ ¼ 0 (cf. [6, p. 197]). From Lemmas 2.2 and 2.3 of [3], we have Lemma 1. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let T ¼ T þ dT 2 BðH1 ; H2 Þ. Then dðRðT Þ; RðT ÞÞ 6 kT þ kkdT k;
dðKer T ; Ker T Þ 6 kT þ kkdT k;
ð2:1Þ
where RðT Þ is the closure of RðT Þ. Lemma 2 [4, Theorem 1]. Let T ¼ T þ dT , T 2 BðH1 ; H2 Þ with the generalized þ inverse T þ and kT þ kkdT k < 1. If RðT Þ \ RðT Þ? ¼ 0, then T exists and þ
kT k 6
kT þ k : 1 kT þ kkdT k
Lemma 3. Let M1 ; M2 be two subspaces of H and P1 ; P2 be two orthogonal projections of H onto M 1 ; M 2 respectively with kðI P1 ÞP2 k 6 d < 1. Suppose that dim M1 ¼ dim M2 < þ1 or M 2 \ M1? ¼ 0. Then kP1 P2 k ¼ kðI P2 ÞP1 k ¼ kðI P1 ÞP2 k: Proof. Define a continuous linear operator A of M 1 to M 2 by Ax ¼ P2 x, x 2 RðT Þ. Then from the assumption kðI P2 ÞP1 k 6 d < 1, we get that kAxk P ð1 dÞkxk;
ð2:2Þ
2 M 1:
Thus Ker A ¼ 0 and RðAÞ is closed in M 2 by Eq. (2.2). So if dim M 1 ¼ dim M 2 < þ1, then RðAÞ ¼ M 2 . Now let x 2 M 2 and ? x 2 RðAÞ . Then ðx; AyÞ ¼ 0 for all y 2 M 1 so that ðx; yÞ ¼ ðx; AyÞ ¼ 0, i.e., ? x 2 M 2 \ M 1 ¼ 0. Since RðAÞ is closed, RðAÞ ¼ M 2 so that by [6, p. 56, Theorem 6.34(i)], kP2 P1 k ¼ kðI P2 ÞP1 k ¼ kðI P1 ÞP2 k 6 d:
768
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
Now we present our main result as follows. Theorem 4. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. Then following conditions are equivalent: (1) (2) (3) (4) (5)
?
RðT Þ \ RðT Þ ¼ 0, ðI þ dTT þ Þ1 T maps Ker T into RðT Þ, Ker T \ ðKer T Þ? ¼ 0, kP P k < 1, kQ Qk < 1,
where P (resp. P ) is the orthogonal projection of Ker T (resp. Ker T ) and Q (resp. Q) is the orthogonal projection of RðT Þ (resp. RðT Þ). ?
Proof. ð1Þ ) ð2Þ. Since Ker T þ ¼ RðT Þ , we get that by [3, Propositions 3.1 and 1 3.2], ðI þ dTT þ Þ T maps Ker T into RðT Þ, i.e., ðI TT þ ÞðI þ dTT þ Þ1 T ðI T þ T Þ ¼ 0
ð2:3Þ
when RðT Þ \ RðT Þ? ¼ 0. ð2Þ ) ð1Þ. Assume that Eq. (2.3) holds. Noting that RðT Þ? ¼ Ker T ¼ Ker T þ , we need to prove that if T þ x ¼ 0 and x ¼ T y for some y 2 H1 ; then x ¼ 0. Put y1 ¼ T þ Ty; y2 ¼ ðI T þ T Þy. Then from the identity T þ T y ¼ 0, we get 1 that y1 þ T þ dTy1 þ T þ dTy2 ¼ 0 so that y1 ¼ ðI þ T þ dT Þ T þ dTy2 . Thus x ¼ ðT þ dT Þy1 þ ðT þ dT Þy2 1
¼ ðT þ dT ÞðI þ T þ dT Þ T þ dTy2 þ dTy2 1
¼ ðI þ dTT þ ÞTT þ ðI þ dTT þ Þ dTy2 þ dTy2 ¼ ðI þ dTT þ ÞðI TT þ ÞðI þ dTT þ Þ1 dT ðI T þ T Þy2 ¼ 0 by Eq. (2.3), that is, T is the stable perturbation of T . ð2Þ ) ð3Þ. Eq. (2.3) implies that
þ 1
ðI T þ T ÞðI þ ðdT Þ ðT Þ Þ ðdT Þ ðI TT þ Þ ¼ 0: ?
þ
ð2:4Þ ?
þ
Since Ker T ¼ RðT Þ ¼ ðI TT ÞH2 and RðT Þ ¼ ðKer T Þ ¼ ðT T ÞH1 , it ? follows from the proof of ð2Þ ) ð1Þ that RðT Þ \ RðT Þ ¼ 0, that is, Ker T \ ? ðKer T Þ ¼ 0. ð3Þ ) ð2Þ. We have RðT Þ \ RðT Þ? ¼ 0 by the assumption. So
ðI T ðT Þþ ÞðI þ ðdT Þ ðT Þþ Þ1 T ðI ðT Þþ T Þ ¼ 0: þ
þ
þ
þ
ð2:5Þ
Noting that ðT Þ ¼ ðT Þ and ðT Þ ¼ ðT Þ , we obtain that from Eq. (2.5) ðI TT þ ÞT ðI þ T þ dT Þ1 ðI T þ T Þ ¼ 0
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
769
so that 1
ðI TT þ ÞðI þ T þ dT Þ T ðI T þ T Þ ¼ 0: þ
ð1Þ ) ð5Þ. In this case, T exists by Lemma 2 and dðRðT Þ; RðT ÞÞ 6 kT kkdT k < 1 by Lemma 1. þ Since dðRðT Þ; RðT ÞÞ P distðx; RðT ÞÞ ¼ kðI T T Þxk when x 2 RðT Þ with kxk ¼ 1, it follows from Lemma 1 that þ
þ
kðI T T ÞTT þ k 6 dðRðT Þ;
RðT ÞÞ 6 kT þ kkdT k < 1:
So by Lemma 3, þ
þ
kTT þ T T k ¼ kðI T T ÞTT þ k 6 kT þ kkdT k < 1 when dim RðT Þ ¼ dim RðT Þ < þ1 or RðT Þ \ RðT Þ? ¼ 0. ð5Þ ) ð1Þ. It is easy to verify that kQ Qk < 1 indicates that RðT Þ \ ? RðT Þ ¼ 0. þ ð3Þ ) ð4Þ. The equivalence of (1) and (3) means that T exists. So by Lemma 1, þ
kðI T þ T ÞT T k 6 dðKer T ; Ker T Þ 6 kT þ kkdT k < 1
ð2:6Þ
and hence þ
þ
kT T T þ T k 6 kðI T þ T ÞT T k 6 kT þ kkdT k < 1
ð2:7Þ ?
by Lemma 3 when dim Ker T ¼ dim Ker T < þ1 or Ker T \ ðKer T Þ ¼ 0. ð4Þ ) ð3Þ is obvious. 1
Remark 1. The condition that ðI þ dTT þ Þ T maps Ker T into RðT Þ guarantees that T has a generalized inverse under sense of Banach spaces. This famous result was given by Nashed (cf. [7]). Combining authorsÕ early work [1,3] with above theorem, we may find that the notation of the stable perturbation of operators on Hilbert spaces not only generalizes the concepts of the rankpreserving perturbation of matrices and the acute perturbation of matrices (cf. [8]) but also characterizes the continuity of the generalized inverses in Hilbert spaces (cf. [3, Corollaries 3.1 and 4.1]). Corollary 5. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. If T and T satisfy one of following conditions, then T is the stable perturbation of T : (1) (2) (3) (4)
RðT Þ ¼ RðT Þ, Ker T ¼ Ker T , dim Ker T ¼ dim Ker T < þ1, dim RðT Þ ¼ dim RðT Þ < þ1.
770
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772 ?
Proof. Obviously, RðT Þ \ RðT Þ ¼ 0 by Theorem 4 when the condition (1) or (2) is satisfied. Now suppose that T and T satisfy condition (3) or (4). Then by the proof of Theorem 4, we have RðT Þ \ RðT Þ? ¼ 0. Corollary 6. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. If T is the stable perturbation of T , then þ
þ
kT T T þ T k 6 minfkT kkdT k; kT þ kkdT kg; þ
þ
kT T TT þ k 6 minfkT kkdT k; kT þ kkdT kg: þ
Proof. If kT k P kT þ k, the statements have been obtained by Eqs. (2.6) and þ (2.7). So we assume that kT k 6 kT þ k. Thus by Lemma 1, dðKer T ; Ker T Þ 6 þ kT kkdT k < 1 and þ
þ
kT T ðI T þ T Þk 6 dðKer T ; Ker T Þ 6 kT kkdT k < 1: Therefore, by Lemma 3 and Theorem 4, we get that þ
þ
kT T T þ T k ¼ kðI T þ T Þ ðI T T Þk þ
þ
¼ kT T ðI T þ T Þk 6 kT kkdT k: Similarly, we may prove the remain.
3. Perturbation analysis þ
In this section, we will give an error estimate for kT T þ k and establish a perturbation analysis for the Eq. (1.2) by means of Corollary 6. Proposition 7. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let ? T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. Suppose that RðT Þ \ RðT Þ ¼ 0. þ Then T exists and pffiffiffi 1þ 5 þ þ þ kT kkT þ kkdT k: kT T k 6 2 Proof. In this case, T þ
þ
þ
exists by Lemma 2. From the identity þ
þ
T T þ ¼ T dTT þ þ T ðI TT þ Þ þ ðI T T ÞT þ ;
ð3:1Þ
we get that by applying the orthogonality of the operators on the right side of Eq. (3.1),
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772 þ
þ
2
þ
771 þ
2
2
kðT T þ Þxk ¼ k T dTT þ x þ T ðI TT þ Þxk þ kðI T T ÞT þ xk ; where x 2 H1 with kxk ¼ 1. Set sin / ¼ kTT þ xk and cos / ¼ kðI TT þ Þxk. Then we have by Corollary 6 þ
þ
þ
þ
kðT T þ Þxk2 ¼ k T dTT þ ðTT þ xÞ þ T ðT T TT þ ÞðI TT þ Þxk2 þ
þ kðT þ T T T ÞT þ ðTT þ xÞk2 þ 2
2
2
2
6 kT k kdT k kT þ k ½ðsin / þ cos /Þ þ sin2 / 1 þ 2 2 2 ¼ kT k kdT k kT þ k ð3 þ 2 sin 2/ cos 2/Þ 2 pffiffiffi 1 þ 2 2 2 6 kT k kdT k kT þ k 3 þ 5 2 pffiffi þ þ þ and hence kT T k 6 1þ2 5 kT kkT þ kkdT k. Remark 2. Proposition 7 is well-known when H1 ; H2 are all finite-dimensional þ (cf. [9]). Since kT k 6 kT þ k=ð1 kT þ kkdT kÞ by Lemma 2 when kT þ kkdT k < 1 ? and RðT Þ \ RðT Þ ¼ 0, it follows from Proposition 1 that pffiffiffi þ kT T þ k 1 þ 5 kT þ kkdT k 6 : ð3:2Þ kT þ k 2 1 kT þ kkdT k Eq. (3.2) improves [4, Theorem 2]. Now we consider the problem (1.2). The unique optimal solution to (1.2) is given by x ¼ T þ b þ ðI T þ T Þp:
ð3:3Þ þ
When p ¼ 0 in Eq. (3.3), x ¼ T b is the minimal norm least squares solution of Eq. (1.2). Suppose that the problem (1.2) is perturbed to min k p yk subject to kT y bk ¼ min kT z bk; z2H1
ð3:4Þ
b ¼ b þ db 2 H2 and T ¼ T þ dT 2 BðH1 ; H2 Þ. Eq. (3.4) where p ¼ p þ dp 2 H1 , has the unique optimal solution þ þ x ¼ T p b þ ðI T T Þ
ð3:5Þ
when RðT Þ is closed in H2 . Set j ¼ kT kkT þ k, T ¼ kdT k=kT k, p ¼ kdpk=kpk and b ¼ kdbk=kbk. Proposition 8. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdk < 1 Assume that RðT Þ \ RðT Þ? ¼ 0. Then
772
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
kx xk j kb Txk kbk 6 jT þ b T þ kxk 1 jT kT kkxk kT kkxk kx pk kpk jT þ p : þ kxk kxk Proof. By Eqs. (3.3) and (3.4), þ
þ
þ
þ
x x ¼ ðT T þ Þðb TpÞ þ T db T dTp þ ðI T T Þdp: Since T þ ðb TpÞ ¼ x p and ðI TT þ Þðb TpÞ ¼ b Tx, it follows from Eq. (3.1) that þ
þ
þ
kx xk 6 kT kkdT kkxk þ kT kkT T TT þ kkb Txk þ
þ
þ kT þ T T T kkx pk þ kT kkdbk þ kdpk: þ
Noting that kT k 6 kT þ k=ð1 kT þ kkdT kÞ and þ
þ
kT T T þ T k 6 kT þ kkdT k;
kT T TT þ k 6 kT þ kkdT k ?
by Corollary 6 when RðT Þ \ RðT Þ ¼ 0, we have kx xk j kb Txk kbk 6 jT þ b T þ kxk 1 jT kT kkxk kT kkxk kx pk kpk jT þ p : þ kxk kxk
References [1] G. Chen, M. Wei, Y. Xue, Perturbation Analysis of the Least Squares Solution in Hilbert Spaces, Linear Algebra Appl. 244 (1996) 69–80. [2] G. Chen, Y. Wei, Perturbation analysis for the projection of a point onto an affine set in Hilbert spaces (Chinese), Chinese Anal. Math. 19 (Ser. A: 4) (1998) 405–410. [3] G. Chen, Y. Xue, Perturbation Analysis for the Operator Equation Tx ¼ b in Banach spaces, J. Math. Anal. Appl. 212 (1997) 107–125. [4] G. Chen, Y. Xue, The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces, Linear Algebra Appl. 285 (1998) 1–6. [5] J. Ding, Perturbation results for projecting a point onto a linear manifold, SIAM J. Matrix Anal. Appl. 19 (3) (1998) 696–700. [6] K. 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: M.Z. Nashed (Ed.), Generalized Inverses and Applications, Academic Press, New York/San Francisco/London, 1976. [8] G.W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev. 19 (4) (1977) 635–662. [9] G.W. Stewart, J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.
Some equivalent conditions of stable perturbation of operators in Hilbert spaces q Yifeng Xue
a,*
, Guoliang Chen
b
a
b
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China Department of Mathematics, East China Normal University, Shanghai 200062, PR China
Abstract Let H1 ; H2 be two Hilbert spaces over the complex field C and let T ; T ¼ T þ dT : H1 ! H2 be two bounded linear operators with the M–P generalized inverse T þ . If RðT Þ \ RðT Þ? ¼ 0, we say that T is the stable perturbation of T . In this paper, we give five equivalent conditions that make T being the stable perturbation of T under the assumption kT þ kkdT k < 1. These equivalent conditions generalize not only the notation of rank-preserving perturbations of matrices but also the notation of acute perturbations of matrices. As a result, we obtain the following: Suppose that RðT Þ \ RðT Þ? ¼ 0 and kT þ kkdT k < 1. Then pffiffiffi 1þ 5 þ þ kT kkT þ kkdT k: kT T þ k 6 2 This result generalizes corresponding result when H1 ; H2 are all finite-dimensional. Ó 2003 Elsevier Inc. All rights reserved. Keywords: M–P generalized inverses; Stable perturbation of operators
q
Supported by the Natural Sciences Foundation, PR China. Corresponding author. E-mail address: [email protected] (Y. Xue).
*
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00810-X
766
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
1. Introduction Let ðH1 ; k kÞ; ðH2 ; k kÞ be two Hilbert spaces over the complex field C and let BðH1 ; H2 Þ denote the Banach space of all bounded linear operators T : H1 ! H2 with the norm kT k ¼ supfkTxk j kxk ¼ 1; x 2 H1 g: For T 2 BðH1 ; H2 Þ, Ker T (resp. RðT Þ) denotes the null space (resp. range) of T . According to [7], T 2 BðH1 ; H2 Þ with RðT Þ closed has a Moore–Penrose inverse T þ , namely, T þ is the unique solution for the equations T þ TT þ ¼ T þ ;
TT þ T ¼ T ;
T þT ¼ I P ;
TT þ ¼ Q;
ð1:1Þ
where P (resp. Q) is the orthogonal projection of H1 onto KerðT Þ (resp. H2 onto þ RðT Þ). We have ðT Þ ¼ ðT þ Þ , here T is the adjoint of T . In [1], Chen et al. proposed two notations about the perturbation of operators on Hilbert spaces, that is, for T ; T ¼ T þ dT 2 BðH1 ; H2 Þ, if RðT Þ \ ? ? RðT Þ ¼ 0 (resp. Ker T \ ðKer T Þ ¼ 0), then T is said to be the type I (resp. Type II) perturbation of T . They are the same when RðT Þ is closed and kT þ kkdT k < 12 (cf. [1, Corollary 3.2]). Later, Chen and Xue introduced a notation so-called the stable perturbation of operators on Banach spaces in [3], that is, if RðT Þ \ Ker T þ ¼ 0, then T is said to be the stable perturbation of T . But ? noting that Ker T þ ¼ ðI TT þ ÞH1 ¼ RðT Þ , we get that T is the stable perturbation of T iff T is the Type I perturbation of T . In this paper, authors will continue to investigate the conditions that make the perturbation of operators stable in Hilbert spaces. As a result, we get ? that if T ; T ¼ T þ dT 2 BðH1 ; H2 Þ with RðT Þ closed, RðT Þ \ RðT Þ ¼ 0 and þ þ kT kkdT k < 1, then T exists and pffiffiffi þ 1 þ kT T þ k 6 1 þ 5 kT kkT þ kkdT k: 2 This result generalized corresponding result in perturbation analysis of matrices with rank-preserving (cf. [9]). Finally, we consider the perturbation problem of the orthogonal projection of a point onto a linear manifold: min kp xk
subject to kTx bk ¼ min kTz bk; z2H1
ð1:2Þ
where p; x 2 H1 , b 2 H2 and T 2 BðH1 ; H2 Þ with RðT Þ closed. The perturbation analysis of Eq. (1.2) was discussed by Ding in [5] when H1 ; H2 are finitedimensional. When H1 ; H2 are all infinite-dimensional, the perturbation analysis of Eq. (1.2) was considered by Chen and Wei in [2], but their result is not good. So we will give a better estimate of the perturbation analysis of Eq. (1.2) than that of [2].
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
767
2. The main result Let M1 ; M2 be two closed subspaces in the Hilbert space H (with the norm k k). Set dðM1 ; M2 Þ ¼ supfdistðx; M2 Þ j kxk ¼ 1; x 2 M1 g: If M1 ¼ 0, we set dðM1 ; M2 Þ ¼ 0 (cf. [6, p. 197]). From Lemmas 2.2 and 2.3 of [3], we have Lemma 1. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let T ¼ T þ dT 2 BðH1 ; H2 Þ. Then dðRðT Þ; RðT ÞÞ 6 kT þ kkdT k;
dðKer T ; Ker T Þ 6 kT þ kkdT k;
ð2:1Þ
where RðT Þ is the closure of RðT Þ. Lemma 2 [4, Theorem 1]. Let T ¼ T þ dT , T 2 BðH1 ; H2 Þ with the generalized þ inverse T þ and kT þ kkdT k < 1. If RðT Þ \ RðT Þ? ¼ 0, then T exists and þ
kT k 6
kT þ k : 1 kT þ kkdT k
Lemma 3. Let M1 ; M2 be two subspaces of H and P1 ; P2 be two orthogonal projections of H onto M 1 ; M 2 respectively with kðI P1 ÞP2 k 6 d < 1. Suppose that dim M1 ¼ dim M2 < þ1 or M 2 \ M1? ¼ 0. Then kP1 P2 k ¼ kðI P2 ÞP1 k ¼ kðI P1 ÞP2 k: Proof. Define a continuous linear operator A of M 1 to M 2 by Ax ¼ P2 x, x 2 RðT Þ. Then from the assumption kðI P2 ÞP1 k 6 d < 1, we get that kAxk P ð1 dÞkxk;
ð2:2Þ
2 M 1:
Thus Ker A ¼ 0 and RðAÞ is closed in M 2 by Eq. (2.2). So if dim M 1 ¼ dim M 2 < þ1, then RðAÞ ¼ M 2 . Now let x 2 M 2 and ? x 2 RðAÞ . Then ðx; AyÞ ¼ 0 for all y 2 M 1 so that ðx; yÞ ¼ ðx; AyÞ ¼ 0, i.e., ? x 2 M 2 \ M 1 ¼ 0. Since RðAÞ is closed, RðAÞ ¼ M 2 so that by [6, p. 56, Theorem 6.34(i)], kP2 P1 k ¼ kðI P2 ÞP1 k ¼ kðI P1 ÞP2 k 6 d:
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Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
Now we present our main result as follows. Theorem 4. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. Then following conditions are equivalent: (1) (2) (3) (4) (5)
?
RðT Þ \ RðT Þ ¼ 0, ðI þ dTT þ Þ1 T maps Ker T into RðT Þ, Ker T \ ðKer T Þ? ¼ 0, kP P k < 1, kQ Qk < 1,
where P (resp. P ) is the orthogonal projection of Ker T (resp. Ker T ) and Q (resp. Q) is the orthogonal projection of RðT Þ (resp. RðT Þ). ?
Proof. ð1Þ ) ð2Þ. Since Ker T þ ¼ RðT Þ , we get that by [3, Propositions 3.1 and 1 3.2], ðI þ dTT þ Þ T maps Ker T into RðT Þ, i.e., ðI TT þ ÞðI þ dTT þ Þ1 T ðI T þ T Þ ¼ 0
ð2:3Þ
when RðT Þ \ RðT Þ? ¼ 0. ð2Þ ) ð1Þ. Assume that Eq. (2.3) holds. Noting that RðT Þ? ¼ Ker T ¼ Ker T þ , we need to prove that if T þ x ¼ 0 and x ¼ T y for some y 2 H1 ; then x ¼ 0. Put y1 ¼ T þ Ty; y2 ¼ ðI T þ T Þy. Then from the identity T þ T y ¼ 0, we get 1 that y1 þ T þ dTy1 þ T þ dTy2 ¼ 0 so that y1 ¼ ðI þ T þ dT Þ T þ dTy2 . Thus x ¼ ðT þ dT Þy1 þ ðT þ dT Þy2 1
¼ ðT þ dT ÞðI þ T þ dT Þ T þ dTy2 þ dTy2 1
¼ ðI þ dTT þ ÞTT þ ðI þ dTT þ Þ dTy2 þ dTy2 ¼ ðI þ dTT þ ÞðI TT þ ÞðI þ dTT þ Þ1 dT ðI T þ T Þy2 ¼ 0 by Eq. (2.3), that is, T is the stable perturbation of T . ð2Þ ) ð3Þ. Eq. (2.3) implies that
þ 1
ðI T þ T ÞðI þ ðdT Þ ðT Þ Þ ðdT Þ ðI TT þ Þ ¼ 0: ?
þ
ð2:4Þ ?
þ
Since Ker T ¼ RðT Þ ¼ ðI TT ÞH2 and RðT Þ ¼ ðKer T Þ ¼ ðT T ÞH1 , it ? follows from the proof of ð2Þ ) ð1Þ that RðT Þ \ RðT Þ ¼ 0, that is, Ker T \ ? ðKer T Þ ¼ 0. ð3Þ ) ð2Þ. We have RðT Þ \ RðT Þ? ¼ 0 by the assumption. So
ðI T ðT Þþ ÞðI þ ðdT Þ ðT Þþ Þ1 T ðI ðT Þþ T Þ ¼ 0: þ
þ
þ
þ
ð2:5Þ
Noting that ðT Þ ¼ ðT Þ and ðT Þ ¼ ðT Þ , we obtain that from Eq. (2.5) ðI TT þ ÞT ðI þ T þ dT Þ1 ðI T þ T Þ ¼ 0
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
769
so that 1
ðI TT þ ÞðI þ T þ dT Þ T ðI T þ T Þ ¼ 0: þ
ð1Þ ) ð5Þ. In this case, T exists by Lemma 2 and dðRðT Þ; RðT ÞÞ 6 kT kkdT k < 1 by Lemma 1. þ Since dðRðT Þ; RðT ÞÞ P distðx; RðT ÞÞ ¼ kðI T T Þxk when x 2 RðT Þ with kxk ¼ 1, it follows from Lemma 1 that þ
þ
kðI T T ÞTT þ k 6 dðRðT Þ;
RðT ÞÞ 6 kT þ kkdT k < 1:
So by Lemma 3, þ
þ
kTT þ T T k ¼ kðI T T ÞTT þ k 6 kT þ kkdT k < 1 when dim RðT Þ ¼ dim RðT Þ < þ1 or RðT Þ \ RðT Þ? ¼ 0. ð5Þ ) ð1Þ. It is easy to verify that kQ Qk < 1 indicates that RðT Þ \ ? RðT Þ ¼ 0. þ ð3Þ ) ð4Þ. The equivalence of (1) and (3) means that T exists. So by Lemma 1, þ
kðI T þ T ÞT T k 6 dðKer T ; Ker T Þ 6 kT þ kkdT k < 1
ð2:6Þ
and hence þ
þ
kT T T þ T k 6 kðI T þ T ÞT T k 6 kT þ kkdT k < 1
ð2:7Þ ?
by Lemma 3 when dim Ker T ¼ dim Ker T < þ1 or Ker T \ ðKer T Þ ¼ 0. ð4Þ ) ð3Þ is obvious. 1
Remark 1. The condition that ðI þ dTT þ Þ T maps Ker T into RðT Þ guarantees that T has a generalized inverse under sense of Banach spaces. This famous result was given by Nashed (cf. [7]). Combining authorsÕ early work [1,3] with above theorem, we may find that the notation of the stable perturbation of operators on Hilbert spaces not only generalizes the concepts of the rankpreserving perturbation of matrices and the acute perturbation of matrices (cf. [8]) but also characterizes the continuity of the generalized inverses in Hilbert spaces (cf. [3, Corollaries 3.1 and 4.1]). Corollary 5. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. If T and T satisfy one of following conditions, then T is the stable perturbation of T : (1) (2) (3) (4)
RðT Þ ¼ RðT Þ, Ker T ¼ Ker T , dim Ker T ¼ dim Ker T < þ1, dim RðT Þ ¼ dim RðT Þ < þ1.
770
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772 ?
Proof. Obviously, RðT Þ \ RðT Þ ¼ 0 by Theorem 4 when the condition (1) or (2) is satisfied. Now suppose that T and T satisfy condition (3) or (4). Then by the proof of Theorem 4, we have RðT Þ \ RðT Þ? ¼ 0. Corollary 6. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. If T is the stable perturbation of T , then þ
þ
kT T T þ T k 6 minfkT kkdT k; kT þ kkdT kg; þ
þ
kT T TT þ k 6 minfkT kkdT k; kT þ kkdT kg: þ
Proof. If kT k P kT þ k, the statements have been obtained by Eqs. (2.6) and þ (2.7). So we assume that kT k 6 kT þ k. Thus by Lemma 1, dðKer T ; Ker T Þ 6 þ kT kkdT k < 1 and þ
þ
kT T ðI T þ T Þk 6 dðKer T ; Ker T Þ 6 kT kkdT k < 1: Therefore, by Lemma 3 and Theorem 4, we get that þ
þ
kT T T þ T k ¼ kðI T þ T Þ ðI T T Þk þ
þ
¼ kT T ðI T þ T Þk 6 kT kkdT k: Similarly, we may prove the remain.
3. Perturbation analysis þ
In this section, we will give an error estimate for kT T þ k and establish a perturbation analysis for the Eq. (1.2) by means of Corollary 6. Proposition 7. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let ? T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdT k < 1. Suppose that RðT Þ \ RðT Þ ¼ 0. þ Then T exists and pffiffiffi 1þ 5 þ þ þ kT kkT þ kkdT k: kT T k 6 2 Proof. In this case, T þ
þ
þ
exists by Lemma 2. From the identity þ
þ
T T þ ¼ T dTT þ þ T ðI TT þ Þ þ ðI T T ÞT þ ;
ð3:1Þ
we get that by applying the orthogonality of the operators on the right side of Eq. (3.1),
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772 þ
þ
2
þ
771 þ
2
2
kðT T þ Þxk ¼ k T dTT þ x þ T ðI TT þ Þxk þ kðI T T ÞT þ xk ; where x 2 H1 with kxk ¼ 1. Set sin / ¼ kTT þ xk and cos / ¼ kðI TT þ Þxk. Then we have by Corollary 6 þ
þ
þ
þ
kðT T þ Þxk2 ¼ k T dTT þ ðTT þ xÞ þ T ðT T TT þ ÞðI TT þ Þxk2 þ
þ kðT þ T T T ÞT þ ðTT þ xÞk2 þ 2
2
2
2
6 kT k kdT k kT þ k ½ðsin / þ cos /Þ þ sin2 / 1 þ 2 2 2 ¼ kT k kdT k kT þ k ð3 þ 2 sin 2/ cos 2/Þ 2 pffiffiffi 1 þ 2 2 2 6 kT k kdT k kT þ k 3 þ 5 2 pffiffi þ þ þ and hence kT T k 6 1þ2 5 kT kkT þ kkdT k. Remark 2. Proposition 7 is well-known when H1 ; H2 are all finite-dimensional þ (cf. [9]). Since kT k 6 kT þ k=ð1 kT þ kkdT kÞ by Lemma 2 when kT þ kkdT k < 1 ? and RðT Þ \ RðT Þ ¼ 0, it follows from Proposition 1 that pffiffiffi þ kT T þ k 1 þ 5 kT þ kkdT k 6 : ð3:2Þ kT þ k 2 1 kT þ kkdT k Eq. (3.2) improves [4, Theorem 2]. Now we consider the problem (1.2). The unique optimal solution to (1.2) is given by x ¼ T þ b þ ðI T þ T Þp:
ð3:3Þ þ
When p ¼ 0 in Eq. (3.3), x ¼ T b is the minimal norm least squares solution of Eq. (1.2). Suppose that the problem (1.2) is perturbed to min k p yk subject to kT y bk ¼ min kT z bk; z2H1
ð3:4Þ
b ¼ b þ db 2 H2 and T ¼ T þ dT 2 BðH1 ; H2 Þ. Eq. (3.4) where p ¼ p þ dp 2 H1 , has the unique optimal solution þ þ x ¼ T p b þ ðI T T Þ
ð3:5Þ
when RðT Þ is closed in H2 . Set j ¼ kT kkT þ k, T ¼ kdT k=kT k, p ¼ kdpk=kpk and b ¼ kdbk=kbk. Proposition 8. Let T 2 BðH1 ; H2 Þ with the generalized inverse T þ and let T ¼ T þ dT 2 BðH1 ; H2 Þ with kT þ kkdk < 1 Assume that RðT Þ \ RðT Þ? ¼ 0. Then
772
Y. Xue, G. Chen / Appl. Math. Comput. 147 (2004) 765–772
kx xk j kb Txk kbk 6 jT þ b T þ kxk 1 jT kT kkxk kT kkxk kx pk kpk jT þ p : þ kxk kxk Proof. By Eqs. (3.3) and (3.4), þ
þ
þ
þ
x x ¼ ðT T þ Þðb TpÞ þ T db T dTp þ ðI T T Þdp: Since T þ ðb TpÞ ¼ x p and ðI TT þ Þðb TpÞ ¼ b Tx, it follows from Eq. (3.1) that þ
þ
þ
kx xk 6 kT kkdT kkxk þ kT kkT T TT þ kkb Txk þ
þ
þ kT þ T T T kkx pk þ kT kkdbk þ kdpk: þ
Noting that kT k 6 kT þ k=ð1 kT þ kkdT kÞ and þ
þ
kT T T þ T k 6 kT þ kkdT k;
kT T TT þ k 6 kT þ kkdT k ?
by Corollary 6 when RðT Þ \ RðT Þ ¼ 0, we have kx xk j kb Txk kbk 6 jT þ b T þ kxk 1 jT kT kkxk kT kkxk kx pk kpk jT þ p : þ kxk kxk
References [1] G. Chen, M. Wei, Y. Xue, Perturbation Analysis of the Least Squares Solution in Hilbert Spaces, Linear Algebra Appl. 244 (1996) 69–80. [2] G. Chen, Y. Wei, Perturbation analysis for the projection of a point onto an affine set in Hilbert spaces (Chinese), Chinese Anal. Math. 19 (Ser. A: 4) (1998) 405–410. [3] G. Chen, Y. Xue, Perturbation Analysis for the Operator Equation Tx ¼ b in Banach spaces, J. Math. Anal. Appl. 212 (1997) 107–125. [4] G. Chen, Y. Xue, The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces, Linear Algebra Appl. 285 (1998) 1–6. [5] J. Ding, Perturbation results for projecting a point onto a linear manifold, SIAM J. Matrix Anal. Appl. 19 (3) (1998) 696–700. [6] K. 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: M.Z. Nashed (Ed.), Generalized Inverses and Applications, Academic Press, New York/San Francisco/London, 1976. [8] G.W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev. 19 (4) (1977) 635–662. [9] G.W. Stewart, J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.