Iterative methods for computing generalized inverses and splittings of operators

Applied Mathematics and Computation 208 (2009) 186–188

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Iterative methods for computing generalized inverses and splittings of operators Biljana Nacˇevska Department of Mathematics and Physics, Faculty of Electrical Engineering and Information Technologies, University ‘‘Ss: Cyril and Methodius”, P.O. Box 574, 1000 Skopje, Macedonia

a r t i c l e

i n f o

a b s t r a c t In this paper we consider an iterative method for computing generalized inverses of linear operators, based on splittings of operators. Thus, the result from [Yong-Lin Chen, Xue-Yuan Tan, Computing generalized inverses of matrices by iterative methods based on splitting of matrices, Appl. Math. Comp. 163 (2005) 309–325] is extended to infinite dimensional settings. Ó 2008 Elsevier Inc. All rights reserved.

Keywords: ð2Þ Generalized inverse AT;S Splitting of operators Iterative methods

1. Introduction Let X, Y be Banach spaces. A 2 LðX; YÞ means that A is a linear bounded operator from X to Y, and we shall always assume that A–0. Let T and S be closed subspaces from X and Y respectively. RðAÞ and NðAÞ stands for the range and the kernel of A. If there exists an operator B 2 LðY; XÞ such that BAB = B, RðBÞ ¼ T and NðBÞ ¼ S, then B is an outer inverse of A with the preð2Þ scribed range T and kernel S. This kind of generalized inverse is usually denoted as AT;S . Most of the other generalized inverses ð2Þ can be represented with AT;S , for special cases of T and S. It is of special interest to compute an outer generalized inverse by iterative methods based on splittings of operators. It is well known that for a given closed subspaces S and T of X and Y, ð2Þ respectively, the operator AT;S exists if and only if S, T and A(T) are closed complemented subspaces of X, Y and Y, respectively, ð2Þ ð2Þ A(T)  S = Y, and AjT : T ? A(T) is invertible. Then AT;S is the unique outer generalized inverse of A satisfying RðAT;S Þ ¼ T and ð2Þ NðAT;S Þ ¼ S. The applications of outer generalized inverse can be found in [1,3–5], and many others (see references cited there). There are several iterative methods for computing outer generalized inverses with prescribed range and kernel of Banach space operators [4,7] and for complex matrices [3,6]. In this paper we extend the results from [6] to infinite dimensional settings. We recall some notations and auxiliary results. If X = L  M for some closed subspaces L and M of X, then the projection onto L parallel to M is denoted by P = PL,M. We use q(A) to denote the spectral radius for the operator A 2 LðXÞ. It is known that A converges, i.e limn?1 An = 0, if and only if q(A) < 1. The following results are well known. ð2Þ

Lemma 1.1. Let A 2 LðX; YÞ, and let T, S be given subsets of X and Y respectively. Then AT;S exists if and only if T, A(T) and S are closed complemented subspaces of X, Y and Y, A(T)  S = Y and AjT : T ? A(T) is invertible. In this case we can choose a closed subspace T1 of X such that T  T1 = X, and the following hold: ð2Þ

AAT;S ¼ P AðTÞ;S ;

ð2Þ

AT;S A ¼ P T;T 1 :

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.030

B. Nacˇevska / Applied Mathematics and Computation 208 (2009) 186–188

187

Lemma 1.2. If B 2 LðXÞ, if L and M are closed subspaces of X such that L  M = X, and if PL,M is the projection of X onto L parallel to M, then (i) PL,MB = B if and only if RðBÞ # L; (ii) BPL,M = B if and only if NðBÞ  M.

2. A criterion for the convergence of iterations ð2Þ

In this section we give a criterion for the convergence of iterations for computing the outer inverse AT;S , and we give some corollaries using our main theorem. ð2Þ

Theorem 2.1. Let A 2 LðX; YÞ, and let T, S be subspaces of X and Y such that AT;S exists. Suppose that A = U  V for some operators ð2Þ U; V 2 LðX; YÞ, such that U K;L exists for subspaces K # X and L # Y. Suppose that there exist some right invertible operator B 2 LðY; XÞ. Then the iteration ð2Þ

ð2Þ

X jþ1 ¼ U K;L VX j þ U K;L ;

X 0 2 LðY; XÞ;

ð2:1Þ

ð2Þ

converges to AT;S if and only if the following three conditions hold: ð2Þ

(i) qðU K;L VÞ < 1, (ii) K = T, (iii) L = S.

Proof. Ü: Let the conditions (i)–(iii) hold. Then, because of Lemmas 1.1 and 1.2, we obtain: ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ðI  U T;S VÞAT;S ¼ AT;S  U T;S VAT;S ¼ AT;S  U T;S ðU  AÞAT;S ¼ AT;S  U T;S UAT;S þ U T;S AAT;S ¼ AT;S  AT;S þ U T;S ¼ U T;S ; which is the same as ð2Þ

ð2Þ

ð2Þ

ð2Þ

AT;S ¼ U T;S VAT;S þ U T;S : Now we have

X jþ1  AT;S ¼ U T;S VX j þ U T;S  ðU T;S VAT;S þ U T;S Þ ¼ U T;S VðX j  AT;S Þ ¼    ¼ ðU T;S VÞjþ1 ðX 0  AT;S Þ: ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

Because of (i), it follows that limj!1 ðX jþ1  AT;S Þ ¼ 0. We have just proved that Xj+1, given with (2.1), converges to AT;S for every X 0 2 LðY; XÞ. ): Let the iteration ð2Þ

ð2Þ

X jþ1 ¼ U K;L VX j þ U K;L ð2Þ

converges to AT;S for every X 0 2 LðY; XÞ. Then ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

kAT;S  ðU K;L VAT;S þ U K;L Þk ¼ kAT;S  ðU K;L VX j þ U K;L Þ þ ðU K;L VX j þ U K;L Þ  ðU K;L VAT;S þ U K;L Þk ð2Þ

ð2Þ

ð2Þ

6 kAT;S  X jþ1 k þ kU K;L VkkX j  AT;S k: ð2Þ

Since X j  AT;S ! 0 as j ? 1, it follows that ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

AT;S ¼ U K;L VAT;S þ U K;L ;

ð2:2Þ

or ð2Þ

AT;S ¼ U K;L ðVAT;S þ IÞ;

ð2:3Þ

and we have T # K. Using (2.2) again, we have ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

X jþ1  AT;S ¼ U K;L VX j þ U K;L  ðU K;L VAT;S þ U K;L Þ ¼

ð2Þ ðU K;L VÞjþ1 ðX 0 ð2Þ AT;S ,



ð2:4Þ

ð2Þ AT;S Þ:

ð2:5Þ ð2Þ ðU K;L VÞ

We take X 0 ¼ B þ where B 2 LðY; XÞ is right invertible. It follows that q < 1. This proves the condition (i). ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ Now, using (i) and (2.3) we have that I  U K;L V is invertible and ðI  U K;L VÞAT;S ¼ U K;L or AT;S ¼ ðI  U K;L VÞ1 U K;L . From the last two equations it is obvious that S = L, that is the condition (iii) holds. ð2Þ By now we have A(T)  S = Y = U(K)  S, T # K and qðU K;S VÞ < 1.

B. Nacˇevska / Applied Mathematics and Computation 208 (2009) 186–188

188

ð2Þ

Now we prove that K # T. Let x0 2 T1 (see Lemma 1.1.). We have AT;S Ax0 ¼ P T;T 1 x0 ¼ 0 and consequently ð2Þ ð2Þ Ax0 2 NðAT;S Þ ¼ S. It follows that U K;S Ax0 ¼ 0. Consider the decomposition X = K  K1 (by Lemma 1.1). Now we have ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

PK;K 1 x0 ¼ U K;S Ux0 ¼ U K;S Ux0  U K;S Ax0 ¼ U K;S ðU  AÞx0 ¼ U K;S Vx0 ; ð2Þ

Then, because P K;K 1 is a projection and qðU K;S VÞ < 1, we get P K;K 1 x0 ¼ 0, which is, again, equivalent to x0 2 K1. We have just proved that K # T, so the condition (ii) holds. h One direction of the previous theorem is proved in [4] using splitting of operators. Anyway, present result is a generalization of the result proved within [6] (Theorem 2.1), where the analogous result is proved for complex matrices. ð2Þ

ð2Þ

Corollary 2.2. Let A, T, S, K, L, U, V be the same as in Theorem 2.1 and let AT;S and U K;L exist. Suppose that there exists some left invertible operator C 2 LðY; XÞ. Then the iteration ð2Þ

ð2Þ

X jþ1 ¼ X j VU K;L þ U K;L ;

X 0 2 LðY; XÞ;

ð2Þ

converges to AT;S for any X0, if and only if conditions (i)–(iii) from Theorem 2.1 hold. ð2Þ

ð2Þ

Proof. The proof is just the same as in Theorem 2.1 taking into consideration that qðU K;L VÞ ¼ qðVU K;L Þ h Definition 2.3. Let A 2 LðX; YÞ and let T and S be subspaces of X and Y, respectively, such that there exists the generalized ð2Þ ð2Þ ð2Þ inverse AT;S . Then A = U  V is called a {T, S}-splitting of A if U T;S exists. If qðU T;S VÞ < 1, then A = U  V is a convergent {T, S}splitting. ð2Þ

Remark 2.4. Theorem 2.1 and Corollary 2.2 carry the following information: if the iteration (Xj)j converges to AT;S for every ð2Þ X0 then U T;S have to exist, that is, there exists a {T, S}-splitting for A. Notice that the concept of {T, S}-splitting for complex matrices is introduced in [3], and for operators it is introduced in [4]. Corollary 2.5. The iteration ð2Þ

ð2Þ

xjþ1 ¼ U T;S Vxj þ U T;S ð2Þ

converges to AT;S b for any x0 and for every b if the conditions (i)–(iii) from Theorem 2.1hold. Let X, Y be Hilbert spaces and let A 2 LðX; YÞ have a closed range. By A  we denote the Moore–Penrose inverse of A. The splitting A = U  V is called proper, if RðAÞ ¼ RðUÞ (see [2]). Corollary 2.6. Let X and Y be Hilbert spaces and let A 2 LðX; YÞ have a closed range. If A has a proper splitting A = U  V, then the iteration

X jþ1 ¼ U y VX j þ U y

ðor

X jþ1 ¼ X j VU y þ U y Þ

converges to A  for any X0, if and only if following three conditions hold: (i) q(U V) < 1; (ii) RðAÞ ¼ RðUÞ; (iii) NðAÞ ¼ NðUÞ.

References [1] [2] [3] [4] [5] [6]

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