Dragan S. Djordjević. 1. Motivation

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1 ON DAVIS-AAN-WEINERGER EXTENSION TEOREM Dragan S. Djordjević Abstract. If R = where = we find a pseudo-inverse form of all solutions W = W such that A = R where A = W and R. In this paper we extend well-known results in a finite dimensional setting proved by Dao-Sheng Zheng (SIAM J. Matrix Anal. Appl. 17 (3) (1996) ). Thus a pseudo inverse form of solutions of the Davis-ahan-Weinberger theorem is established. 1. Motivation Let Z denote an arbitrary ilbert space and let and denote closed mutually orthogonal subspaces of Z such that Z =. We use L( ) to denote the set of all bounded operators from into and L() = L( ). For T L( ) let R(T ) and N (T ) respectively denote the range and the kernel of T. Let = L() and L( ) be given operators such that ρ = R where R = : Mathematics Subject Classification. 47A05 47A20 15A09. ey words and phrases. Douglas-ahan-Weinberger theorem More-Penrose inverse. This work is supported by the Ministry of Science Technology and Development of Seriba grant no Typeset by AMS-TEX

2 2 DRAGAN S. DJORDJEVIĆ Notice that R always holds. We consider the following problem. Find an operator W = W L() such that the selfadjoint operator A = : W satisfies the norm condition A = R = ρ. This is a typical selfadjoint dilation problem. We mention that a nonselfadjoint form is also important. The result which is known as the Davis-ahan-Weinberger theorem is proved in 5 Theorem 1.2 and stated as follows: Theorem (DW). Let C satisfy µ C µ and < µ. Then there exists W such that C µ. Indeed those W W which have this property are exactly those of the form W = L + µ(i ) 1/2 Z(I L L) 1/2 where = (µ 2 I ) 1/2 L = (µ 2 ) 1/2 C and Z is an arbitrary contraction. If is compact then W may be chosen compact. The selfadjoint version of the previous theorem follows (see 5 Corollary 1.3): Corollary (DW-SA). Let be selfadjoint and µ and < µ. Then there exists selfadjoint W such that µ. Indeed those W W which have this property are exactly those such that µi + (µi + ) 1 W µi (µi ) 1. The following result is a central solution obtained from Corollary (DW- SA) (see 5 (1.7)). One strightforward proof of this result is given in 15 Lemma 3.1 (although the proof is given for complex matrices a careful reading shows that it is valid for operators on arbitrary ilbert spaces also).

3 PSEUDO INVERSE FORM OF TE DAVIS-AAN-WEINERGER TEOREM3 Corollary (DW-central). Let R = : where = σ R and σ >. If W σ = (σ 2 2 ) 1 and then A σ σ. A σ = A selfadjoint part of this problem is proved by M. G. rein (see 9 and 13 Sec. 125). One special case of the Davis-ahan-Weinberger theorem was proved by. Sz.-Nagy and C. Foias (see 14 Theorem 1 and also 3). Several proofs of Theorem (DW) are presented in 4 Sec. 3 5 Theorem 1.2 and 12 Theorem 1. The boundary case appears if we assume = R = µ. One solution (as a non-selfadjoint extension) is found in 4 Sec. 3. In this case at least one of µi and µi + is not invertible but we can consider W σ their Moore-Penrose inverses (in the case when they exist). Zheng used this idea in 15 Theorem 4.1 and completely solved this problem in finite dimensional settings. ahan also found one solution of this problem but he did not publish his results which appeared in 11 p without any proof. See also results of Fioas and Frazho 6 Chapter IV. Zhang also proved Theorem (DW-central) in finite dimensional settings under the more general assumption µ. Finally we mention that finitedimensional dilation results of this type have lots of applications in numerical analysis (see and 10). In this paper we extend Zheng s results for operators on arbitrary ilbert spaces. 2. Notations We use notations in the same way as in 15.

4 4 DRAGAN S. DJORDJEVIĆ Recall that an operator T L( ) is the Moore-Penrose inverse of T L( ) if the following is satisfied: T T T = T T T T = T (T T ) = T T (T T ) = T T. It is well-known that T exists if and only if R(T ) is closed and in this case T is unique 2. Assume that T L() and 0 is not the point of accumulation of the spectrum σ(t ) of T. If the point {0} is the pole of the resolvent λ (λ T ) 1 then the order of this pole is the Drazin index (or the index) of T denoted by ind(t ). Notice that ind(t ) < holds if and only if there exists the Drazin inverse of T i.e. there exists the unique operator T D L() such that the following hold: T D T T D = T D T T D = T D T T n+1 T D = T n and the least n in the previous definition is equal to ind(t ). If ind(t ) 1 then T D is known as the group inverse of T denoted by T #. If ind(t ) = 0 then T is invertible and T 1 = T D. In this article the group inverse is of special interest. If ind(t ) 1 then = R(T ) + N (T ) and this sum is not necessarily orthogonal. Also T has the matrix form with respect to this decomposition: 0 0 N (T ) T = : 0 T 1 R(T ) N (T ) R(T ) where T 1 = T R(T ) : R(T ) R(T ) is invertible 2. In the case when T is selfadjoint and has a closed range the Moore- Penrose inverse coincides with the group inverse of T. Also R(T ) is closed if and only if 0 is not the accumulation point of σ(t ). decomposition = R(T ) N (T ) is orthogonal. In this case the If T = T L() then we write T 0 if and only if (T x x) 0 for all x where ( ) is the inner product in. Also T > 0 if and only if T 0 and T is invertible.

5 PSEUDO INVERSE FORM OF TE DAVIS-AAN-WEINERGER TEOREM5 3. Results The following result is proved in 1. Lemma 3.1. Let S11 S S = 12 S12 : S 22 where S 11 = S 11 S 22 = S 22 and R(S 11 ) is closed. Then S 0 if and only if the following is satisfied: S 11 0 S 11 S 11 S 12 = S 12 and S 22 S 12S 11 S Although the original proof in 1 is given for finite dimensional spaces and the result is valid in infinite dimensional settings also. We now prove the first auxiliary result. Lemma 3.2. Let R = : = and ρ = R. Then N (ρ ) N () R(ρ ) R( ) N (ρ+) N () and R(ρ+) R( ). Proof. Obviously ρ. Let x N (ρ ) and x = 1. Then ρ 2 Rx 2 = x 2 + x 2 = ρ 2 + x 2 implying x = 0. The rest of the proof is similar. Notice that if there exists any x N (ρ ) and x = 1 then = ρ = R. The following result represents a pseudo inverse form of solutions of the Davis-ahan-Weinberger theorem. Theorem 3.3. Let R = : = ρ = R W = W L() A = and let R(ρ ) and R(ρ+) be closed. Then A = ρ if and only if W (ρ + ) ρ W ρ (ρ ).

6 6 DRAGAN S. DJORDJEVIĆ Proof. Obviously ρ = R A. Since A = A in order to prove A ρ it is enough to prove ρ A 0 and ρ + A 0. Notice that ρ ρ A = ρ W From Lemma 3.1 we know that ρ A 0 if and only if: (1) ρ 0; (2) (ρ )(ρ ) = ; (3) ρ W ( )(ρ ) ( ) 0. We know that (1) always holds. The condition (2) is equivalent to R( ) R(ρ ) which is always true according to Lemma 3.2. Finally (3) is equivalent to ρ (ρ ) W. Similarly ρ + A 0 is equivalent to (ρ + ) ρ W. Now we prove the extension of Corollary (DW-central). Theorem 3.4. Let R = R(ρ + ) be closed. If. = ρ = R and let R(ρ ) and W = (ρ 2 2 ) and then A = W A = ρ = R. Proof. The case ρ = 0 is trivial. ence assume ρ > 0. Since the Moore- Penrose inverse of a selfadjoint operator coincides with its group inverse we conclude that the decomposition = N (ρ ) R(ρ ) is orthogonal and ρ = 0 0 : 0 M N (ρ ) R(ρ ) N (ρ ) R(ρ )

7 PSEUDO INVERSE FORM OF TE DAVIS-AAN-WEINERGER TEOREM7 where M is invertible and M > 0. We conclude that and ρ + have the following matrix forms with respect to the same decomposition of : ρ 0 2ρ 0 = ρ + =. 0 ρ M 0 2ρ M Since ρ+ 0 we conclude 0 < M 2ρ. From ind(ρ+) 1 we conclude that ind(2ρ M) 1. Now R(ρ ) = N (2ρ M) R(2ρ M) and this decomposition is orthogonal since 2ρ M is selfadjoint. Also 0 0 N (2ρ M) N (2ρ M) 2ρ M = : 0 N R(2ρ M) R(2ρ M) where N is invertible. Since M 2ρ we conclude N > 0. Notice that 2ρ 0 M = 0 2ρ N hence from M > 0 we get 0 < N < 2ρ. Finally we get = ρ ρ 0 ρ = ρ N ρ 0 0 2ρ N ρ + = 2ρ N and conclude N (ρ+) = N (2ρ M). From Lemma 3.2 we know that N (ρ ) N () and N (ρ + ) N () implying the following decomposition of : N (ρ ) = : N (ρ + ) R(2ρ M) and also the matrix form of R: ρ 0 0 N (ρ ) 0 ρ 0 R = : N (ρ + ) R(2ρ M) N (ρ ) N (ρ + ) R(2ρ M)

8 8 DRAGAN S. DJORDJEVIĆ where 1 = N ρ. Notice that ρ < 1 < ρ. If P R(2ρ M) is the orthogonal projection from onto R(2ρ M) and P R(2ρ M) projection from onto R(2ρ M) then 1 R 1 = = P R(2ρ M) RP R(2ρ M) implying R 1 R. 1 Let W ρ = 1 1 (ρ ) 1 1 and 1 1 A ρ = 1 W ρ. is the orthogonal From Lemma (DW-central) we know that A ρ R 1 R = ρ. Now we have the matrix form of A: ρ ρ 0 0 A = W ρ It is easy to see that A = ρ. We only have to prove the equality = ρ ρ A ρ (ρ 2 2 ) = 1 1 (ρ ) 1 1. Since ρ and ρ are not accumulation points of the spectrum σ() we conclude that ρ 2 is not the accumulation point of 2. ence (ρ 2 2 ) exists. Now we compute (ρ 2 2 ) = = ρ ρ N 1 (2ρ N) 1 = 1 1 N 1 (2ρ N) 1 1 = 1 1 (ρ ) As a corollary we get the following result which can not be verified easily by a direct computation.

9 PSEUDO INVERSE FORM OF TE DAVIS-AAN-WEINERGER TEOREM9 Corollary 3.5. If R(ρ ) and R(ρ + ) are closed where ρ = R R = and = then (ρ + ) ρ (ρ 2 2 ) ρ (ρ ). Thus we extended Zheng s results in 15 Theorem 4.1 and Theorem 4.2. References 1 A. Albert Conditions for positive and negative definitness in terms of pseudoinverses SIAM J. Appl. Math. 17 (1969) S. R. Caradus Generalized inverses and operator theory Queen s Paper in Pure and Applied Mathematics Queen s University ingston Ontario Z. Ceauşescu and C. Foias On intertwining dilations Acta Sci. Math. (Szeged) 40 (1978) C. Davis An extremal problem for extensions of a sesquilinear form Linear Algebra Appl. 13 (1976) C. Davis W. M. ahan and. F. Weinberger Norm-preserving dilations and their applications SIAM J. Numer. Anal. 19 (3) (1982) C. Foias and A. E. Frazho The commutant lifting approach to interpolation problems Operator Theory: Advances and Applications 44 irkhauser W. ahan Spectra of nearly ermitian matrices Proc. Amer. Math. Soc. 48 (1) (1975) W. ahan. N. Parlett and E. Jiang Residual bounds on approximate eigensystems of nonnormal matrices SIAM J. Numer. Anal. 19 (3) (1982) M. G. rein The theory of selfadjoint extensions of semi-bounded ermitian transformations and its applications Mat. Sb. 20 (1947) ; 21 (1947) J. Meinguet On the Davis-ahan-Weinberger solution of the norm-preserving problem Numer. Math. 49 (1986) N. Parlett The symmetric Eigenvalue Problem Prentice-all Englewood Cliffs NJ S. Parrott On a quotioent norm and the Sz.-Nagy-Foias lifting theorem J. Funct. Anal. 30 (1978) F. Riesz and. Sz-Nagy Lecons d analyse fonctionnelle Akadémiai iadó. udapest Sz-Nagy and C. Foias Forme triangulaire d une contraction et factorisation de la fonction caracteristique Acta Sci. Math (Szeged) 28 (1967) Dao-Sheng Zheng Further study and generalization of ahan s matrix extension theorem SIAM J. Matrix Anal. Appl. 17 (3) (1996) Department of Mathematics Faculty of Sciences University of Nis P.O. ox Nis Serbia dragan@pmf.ni.ac.yu ganedj@eunet.yu