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1 REPRESENTATONS FOR THE DRAZN NVERSE OF BOUNDED OPERATORS ON BANACH SPACE DRAGANA S. CVETKOVĆ-LĆ AND MN WE Abstract. n this paper a representation is given for the Drazin inverse of a 2 2operator matrix extending to Banach spaces results of Hartwig Li and Wei SAM J. Matrix Anal. Appl pp Also formulae are derived for the Drazin inverse of an operator matrix M under some new conditions. Key words. Operator matrix Drazin inverse D-invertibility GD-invertibility. AMS subject classifications. 47A52 47A62 5A24.. ntroduction. Throughout this paper X and are Banach spaces over the same field. We denote the set of all bounded linear operators from X into by BX and by BX whenx =. ForA BX let RA N A σa andra bethe range the null space the spectrum and the spectral radius of A respectively. By X we denote the identity operator on X. n 958 Drazin 6 introduced a pseudoinverse in associative rings and semigroups that now carries his name. When A is an algebra and a Athenb Ais the Drazin inverse of a if. ab = ba b = bab and a ba A nil A nil is the set of all nilpotent elements of algebra A. Caradus 5 King 23 and Lay 25 investigated the Drazin inverse in the setting of bounded linear operators on complex Banach spaces. Caradus 5 proved that a bounded linear operator T on a complex Banach space has the Drazin inverse if and Received by the editors September Accepted for publication October Handling Editor: Bit-Shun Tam. Department of Mathematics Faculty of Sciences and Mathematics University of Niš P.O. Box 224 Višegradska 33 8 Niš Serbia dragana@pmf.ni.ac.rs. Supported by Grant No. 443 of the Ministry of Science Technology and Development Republic of Serbia. nstitute of Mathematics School of Mathematical Science Fudan University Shanghai 2433 P. R. of China and Key Laboratory of Nonlinear Science Fudan University Education of Ministry ymwei@fudan.edu.cn. Supported by the National Natural Science Foundation of China under grant 875 Shanghai Municipal Education Commission Dawn Project and Shanghai Municipal Science and Technology Committee under grant 9DZ22729 and KLMM9. 63

2 64 D. Cvetković-lić and.wei only if is a pole of the resolvent λ T of T. The order of the pole is equal to the Drazin index of T which we shall denote by inda ori A. n this case we say that A is D-invertible. f inda =k then Drazin inverse of A denoted by A D satisfies.2 A k+ A D = A k A D AA D = A D AA D = A D A and k is the smallest integer such that.2 is satisfied. f inda then A D is known as the group inverse of A denoted by A. A is invertible if and only if inda = and in this case A D = A. Harte 2 and Koliha 24 observed that in Banach algebra it is more natural to replace the nilpotent element in. by a quasinilpotent element. n the case when a ba in. is allowed to be quasinilpotent we call b the generalized Drazin inverse g-drazin inverse of a and say that a is GD-invertible. g-drazin inverse was introduced in the paper of Koliha 24 and it has many applications in a number of areas. Harte 2 associated with each quasipolar operator T an operator T whichis an equivalent to the generalized Drazin inverse. Nashed and Zhao 29 investigated the Drazin inverse for closed linear operators and applied it to singular evolution equations and partial differential operators. Drazin 7 investigated extremal definitions of generalized inverses that give a generalization of the original Drazin inverse. Finding an explicit representation for the Drazin inverse of a general 2 2block matrix posed by Campbell in 4 appears to be difficult. This problem was investigated in many papers see n this paper we give a representation for the Drazin inverse of a 2 2 bounded operator matrix. We show that the results given by Hartwig Li and Wei 22 are preserved when passing from matrices to bounded linear operators on a Banach space. Also we derive formulae for the Drazin inverse of an operator matrix M under some new conditions. f/ accσa then the function z fz can be defined as fz =ina neighborhood of and fz =/z in a neighborhood of σa\{}. Function z fz is regular in a neighborhood of σa and the generalized Drazin inverse of A is defined using the functional calculus as A d = fa. An operator A BX is GD-invertible if / accσa and in this case the spectral idempotent P of A corresponding to {} is given by P = AA d see the well-known Koliha s paper 24. f A is GD-invertible then the resolvent function z z A is defined in a punctured neighborhood of {} and the generalized Drazin inverse of A is the operator A d such that A d AA d = A d AA d = A d A and A AA d is quasinilpotent. t is well-known that if A BX is GD-invertible then using the following

3 Drazin nverse of Bounded Operators on Banach Space 65 decomposition we have that A A = : A 2 X = A : NP is invertible and A 2 : is quasinilpotent operator. n this case the generalized Drazin inverse of A has the following matrix decomposition: A A d = :. For other important properties of Drazin inverses see Main results. Firstly we will state a very useful result concerning the additive properties of Drazin inverses which is the main result proved in 6 with a π = aa d. Theorem 2.. Let a b be GD-invertible elements of algebra A such that Then a + b is GD-invertible and a π b = b ab π = a b π aba π =. a + b d = b d + b d n+2 aa + b n a π + b π a d n= n= + b π a + b n ba d n+2 b d n+2 aa + b n ba d n= n= b d aa + b n ba d n+2 b d k+2 aa + b n+k+ ba d n+2. n= k= Next we extend 22 Lemma 2.4 to the linear operator. Lemma 2.2. Let M BX G BX and H B X be operators such that HG = X. f M is GD-invertible operator then the operator GMH is GDinvertible and 2. GMH d = GM d H.

4 66 D. Cvetković-lić and.wei and Proof. tisevidentthat GM d HGMHGM d H=GM d MM d H = GM d H GM d HGMH =GM d MH = GMM d H =GMHGM d H. To prove that GMH GMHGM d H is a quasinilpotent note that GMH GMHGM d H = GM MM d H. Since M MM d is quasinilpotent we have r GMH GMHGM d H = r GM MM d H = lim GM MM d H n n n = lim G M MM d n n H n Hence 2. is valid. lim G n n M MM d n H n =. n From now on we will assume that X and are Banach spaces and Z = X. For A BX B B X C BX andd B consider the operator A B M = BZ. C D Theorem 2.3. f A and D are GD-invertible operators such that BC = and DC = then M is GD-invertible and M d = A d CA d 2 X + D d 2.2 X = XA B D = A d n+2 BD n D π + A π A n BD d n+2 A d BD d n= n= and = CXD d + CA d X.

5 Drazin nverse of Bounded Operators on Banach Space 67 Proof. We rewrite M = P + Q P = 4 Theorem 5. P d is GD-invertible and A P d d X = D d A B D and Q = C. By X = XA B D is defined by 2.2. Also Q is GD-invertible and Q d =. Now we have that the condition P π Q = Q is equivalent to AX + BD d C = D π C = C 2.3 as the condition PQP π = is equivalent to BCA π =DCA π = BCAX + BD d = DCAX + BD d =. 2.4 Since BC =anddc =from2.3 and 2.4 we get that P π Q = Q and PQP π = so by Theorem 2. we have that M is GD-invertible and M d = P d + = = M n QP d n+2 n= A d X D d A d X D d A d = CA d 2 + n A B C D n= + X + D d for = CXD d + CA d X. i= CA d n+2 n+2 CA d i XD d n+2 i i= CA d 2 2 CA d i XD d 2 i Remark. Theorem 2.3 is a strengthening of 4 Theorem 5.3 since it shows that one of the conditions of Theorem 2.3 BD = is actually redundant. Theorem 2.4. f A and D are GD-invertible operators such that 2.5 C AA d B = A AA d B =

6 68 D. Cvetković-lić and.wei and S = D CA d B is nonsingular then M is GD-invertible and ψ " #! ψ " M d AA d B X = + R R + R i+ 2.6 C AA d A i #! 2.7 A d + A d BS CA d A d BS R = S CA d S. Proof. n8itisprovedthatσa σm =σa σd so we conclude that / accσm i.e. M is GD-invertible. Using that X = for P = AA d wehave B = B B 2 Now we have M = : M = 2 = A B A 2 B 2 C C 2 D A B A 2 B 2 C C 2 D A B C D C 2 B 2 A 2 : and C = C C 2 : :. 2 = = : :. Since = 2 using Lemma 2.2 we have that M d = M d 2 so we proceed towards finding the Drazin inverse of M.

7 Drazin nverse of Bounded Operators on Banach Space 69 n order to get an explicit formula for M d we partition M as a 2 2block-matrix i.e. A3 B 3 M = C 3 D 3 A B A 3 = D C B 3 = C 2 C 3 = B 2 D3 = A 2. From 2.5 we get C 2 B 2 =anda 2 B 2 =sob 3 C 3 =andd 3 C 3 =. Alsoby σa 3 σa =σa σd it follows that A 3 is GD-invertible. Applying Theorem 2.3 we get that M d = = A d 3 C 3 A d 3 2 C 3 A d 3 A d 3 A d 3 i+2 B 3 D3 i C 3 A d 3 i+3 B 3 D3 i A d 3 i+ B 3 D3 i For the operator matrix A 3 we have that its upper left block the operator A is nonsingular and its Schur complement SA 3 =D C A B = D CA d B is nonsingular which implies that the operator A 3 is nonsingular and A A + A 3 = B S C A A B S S C A S. Now M d = M d 2 = 3 +. C 3 A d 3 A d A d 3 i+ B 3 D3 i 3 = 4 = : :.

8 62 D. Cvetković-lić and.wei tisobviousthat 4 3 =. Let us denote by R = 3 A d = : 6 = :. Obviously R is given by 2.7. Now M d = Z + 5 C 3 A d 3 4 R Z + 3 A d 3 i+ B 3 D3 i 6. By computation we get that AA 5 C 3 A d d 3 B 4 = R 3 A d 3 i+ B 3 D3 i 6 = 3 A d 3 i 4 3 A d 3B D3 i 6 = R i AA d A i R C AA d = R i+ C AA d A i so 2.6 is valid. Remark 2. Theorem 2.3 generalizes 22 Theorem 3. to the bounded linear operator. Taking conjugate operator of M in Theorem 2.4 we derived the following corollary: Corollary 2.5. f A and D are GD-invertible operators such that C AA d B = C AA d A = and S = D CA d B is nonsingular then M is GD-invertible and M d = + Ai AA d B R i+ R + R C AA d R is defined by 2.7. f an additional condition C AA d A = is satisfied in Theorem 2.4 we get a simpler formula for M d :

9 Drazin nverse of Bounded Operators on Banach Space 62 Corollary 2.6. f A and D are GD-invertible operators such that C AA d B = A AA d B = C AA d A = and S = D CA d B is nonsingular then M is GD-invertible and AA M d d B = + R R + R C AA d R is defined by 2.7. n the paper of Miao 28 a representation of the Drazin inverse of block-matrices M is given under the conditions: C AA D = AA D B = and S = D CA D B =. Hartwig et al. 22 generalized this result in Theorem 4. and gave a representation of the Drazin inverse of block-matrix M under the conditions: C AA D B = A AA D B = and S = D CA D B =. n the following theorem we generalized Theorem 4. from 22 to the linear bounded operator. Theorem 2.7. f A and D are GD-invertible operators such that C AA d B = A AA d B = S = D CA d B = and the operator AW is GD-invertible then M is GD-invertible and 2.8 M d = 2.9 ψ + " AA d B R = # R! R ψ CA d + X R i+ " A dw A d B C AA d A i and A dw =AW d 2 A is the weighted Drazin inverse of A with weight operator W = AA d + A d BCA d. Proof. Using the notations and method from the proof of Theorem 2.4 we have that M d = A d 3 A d 3 i+2 B 3 D3 i C 3 A d 3 2 C 3 A d 3 i+3 B 3 D3 i = C 3 A d 3 A d 3 A d 3 i+ B 3 D3 i. #!

10 622 D. Cvetković-lić and.wei Now prove that the generalized Drazin inverse of A 3 is given by F = C A A H 2 d A A B H = + A B C A. Remark that from the fact that AW is GD-invertible it follows that A H is GD-invertible. By computation we check that A 3 F = FA 3 and FA 3 F = F. To prove that the operator A 3 FA 3 is a quasinilpotent we will use the fact that for bounded operators A and B on Banach spaces rab = rba. First note that A 3 = C A A A B and H = A B C A. Since it follows that A 3 FA 3 = C A A HA H d A A B r A 3 FA 3 = r A HA H d A H = so A 3 FA 3 is a quasinilpotent. Hence A d 3 = F. Now for R = 3 A d 3 4 we get that 2.8 holds. By computation we obtain that R = 3 A d 3 4 = CA d A dw A d B H W = = AA d + A d BCA d. We obtain the following corollary by taking conjugate operator: Corollary 2.8. f A and D are GD-invertible operators such that C AA d B = C AA d A = S = D CA d B = and the operator AW is GD-invertible then M is GD-invertible and A M d i AA d B = + R i+ R + R C AA d R is given by 2.9 in Theorem 2.7. f the condition C AA d A = is added to Theorem 2.7 we have a simpler formula for M d.

11 Drazin nverse of Bounded Operators on Banach Space 623 Corollary 2.9. f A and D are GD-invertible operators such that C AA d B = C AA d B = A AA d B = S = D CA d B = and the operator AW is GD-invertible then M is GD-invertible and M d = AA d B + R is given by 2.9 in Theorem 2.7. R R + R C AA d The next theorem presents new conditions under which we give a representation of M d in terms of the block-operators of M. Theorem 2.. f A and D are GD-invertible operators and 2. AA d B = and C AA d = then M is GD-invertible and M d = R d + CA d + R π R i CA d i+2 A d + R = AA d A B D and R d = AA d A i BD d i+2 D d. Proof. As in the proof of the Theorem 2.4 we conclude that M is GD-invertible. Using that X = for P = AA d wehave B = Now B B 2 M = : A B A 2 B 2 C C 2 D : and C = C C 2 :. M = J 2 MJ A 2 B 2 = C 2 D C : B A

12 624 D. Cvetković-lić and.wei J 2 = :. and J = : Using Lemma 2.2 we deduce that M d = J M dj 2. n order to compute M d it suffices to find the Drazin inverse of M. To derive an explicit formula for M d we partition M as a 2 2 block-matrix i.e. A3 B 3 M = C 3 D 3 Since A2 B 2 A 3 = D C 2 B 3 = C C 3 = B D3 = A. B 3 C 3 = C B = CAA d B = and D 3 C 3 = A B = AA d B =. by 2. we have B 3 C 3 =D 3 C 3 =andb =. Similarly as in the proof of the Theorem 2.4 we conclude that A 3 is GD-invertible operator. Now by Theorem 2.3 M d = Ad 3 A π 3 A i 3B 3 A i+2 A d 3B 3 A A = A d 3 B3 A + A π 3 A i 3B 3 A i+2 + A. By the second condition of 2. we obtain that C 2 = as for the operator A 3 we have that B 2 C 2 = and DC 2 =. Applying Theorem 2.3 to A 3 weget A d 3 = A i 2 B 2D d i+2 D d.

13 Drazin nverse of Bounded Operators on Banach Space 625 Now M d = J M d J 2 = J 3 A d 3 J4 + B 3 A J 5 + J3 A π 3 = R d + J 3 B 3 A J 5 R = J 3 A 3 J 4 J 3 = + R π J 3 J 4 = A A i 3 B 3A d i+2 J 5 + A i 3B 3 A i+2 J 5 + tisevidentthatj 4 J 3 =. By computation we get that J 3 B 3 A J 5 = CA d J 3 A i 3 B 3A i+2 J 5 = R i Also from the definition of R wehavethat R = AA d A B D A d and J 5 =. CA d i+2. and by 4 Theorem 5. R d = AA d A i BD d i+2 D d. 3. Concluding remarks. The whole paper would appear to be valid in general Banach algebras not just algebras of operators. Whenever P = P 2 G for a Banach algebra G there is an induced block structure A M G = N B in which A and B are Banach algebras and M and N are bimodules over A and B.

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