Faculty of Sciences and Mathematics,University of Niˇs,P.O.Box 224,18000 Niˇs,Serbia

E-mail:dijana@pmf.ni.ac.rs

A NOTE ON THE REPRESENTATIONS FOR THE GENERALIZED DRAZIN INVERSE OF BLOCK MATRICES∗

Dijana MOSI´C

Faculty of Sciences and Mathematics,University of Niˇs,P.O.Box 224,18000 Niˇs,Serbia

E-mail:dijana@pmf.ni.ac.rs

generalized Drazin inverse;block matrix;Banach algebra

2010 MR Subject Classifcation46H05;47A05;15A09

1 Introduction

Let A be a complex unital Banach algebra with unit 1.For a∈A,we denote by σ(a)the spectrum of a.The sets of all invertible,nilpotent and quasinilpotent elements(σ(a)={0})of A will be denoted by A-1,Aniland Aqnil,respectively.

Let us recall that the generalized Drazin inverse of a∈A(or Koliha-Drazin inverse of a) is the unique element ad∈A which satisfes

The generalized Drazin inverse adexists if and only if 0/∈acc σ(a)(see[1]).It is well-known that aπ=1-aadis the spectral idempotent of a corresponding to the set{0}.We use Adto denote the set of all generalized Drazin invertible elements of A.

The following result is well-known for matrices[2,Theorem 2.1],for bounded linear operators[3,Theorem 2.3]and for elements of Banach algebra[4].

Lemma 1.1([4,Example 4.5])Let a,b∈Adand let ab=0.Then a+b∈Adand

If p=p2∈A is an idempotent,we can represent element a∈A as

where a11=pap,a12=pa(1-p),a21=(1-p)ap,a22=(1-p)a(1-p).

We present well-known result on the generalized Drazin inverse of a triangular block matrix.

Lemma 1.2([4,Theorem 2.3])Let

relative to the idempotent p∈A.If a∈(pAp)dand b∈((1-p)A(1-p))d,then x∈Adand

We state the auxiliary results which are proved for matrices[5]and Banach space operators [6],and they are equally true for elements of Banach algebras.

Lemma 1.3Let p∈A be an idempotent,b∈pA(1-p)and c∈(1-p)Ap.If bc∈(pAp)d, then cb∈((1-p)A(1-p))d,(cb)d=c[(bc)d]2b and b(cb)d=(bc)db.

2 Results

Let

relative to the idempotent p∈A,a∈(pAp)dand d∈((1-p)A(1-p))d.

We present the main result which involve new formula for the generalized Drazin inverse of x in terms of adand(bc)dunder some conditions.The following result is a generalization of [6,Theorem 3.8]for the generalized Drazin inverse of an anti-triangular operator matrix.

Theorem 2.1Let x be defned as in(2.1)and let bc∈(pAp)d.If

then x∈Adand

where

ProofWe can write

By the assumptions,we obtain that yz=0.

To prove that y∈Ad,let

Observe that y1y2=0.Since aaπ∈(pAp)qnil,(aaπ)d=0.Using Lemma 1.2,we see that y1∈Adand

From bcaπ=bc∈(pAp)dand Lemma 1.4,we deduce that y2∈Adand

Applying Lemma 1.3,note that caπb∈((1-p)A(1-p))d.So,

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and,for n≥0,

Also,for n≥1,

By Lemma 1.1,we deduce that y∈Adand

By Lemma 1.2,we have that z∈Ad,

It follows that zzπ=0 implying z∈A#and z#=zd.Note that z#y1=0.

Using Lemma 1.1,x∈Adand

Notice that the conditions aaπb=0,bcaad=0 and dc=0 of Theorem 2.1 are equivalent with the following geometrical conditions:

where x◦={y∈A:xy=0}.

Applying Theorem 2.1,we obtain the next consequences.

Corollary 2.2Let x be defned as in(2.1),d∈((1-p)A(1-p))d,bc∈(pAp)dand dc=0.

(i)If a∈(pAp)qniland ab=0,then x∈Adand

(ii)If bca=0 and a2=a,then x∈Adand

where

(iii)If a∈(pAp)-1and bc=0,then x∈Adand

(iv)If a∈(pAp)d,aπb=0 and caad=0,then x∈Adand

(v)If a∈(pAp)d,aaπb=0 and bca=0,then x∈Adand

where

Proof(i)It follows by ad=0.

(ii)Since ad=a,we prove this part.

The parts(iii)and(v)follow by direct computations.

(iv)From aπb=0 and caad=0,we get cad=0,caπ=c,caad=0,(bc)2=bcaπbc=0, cb=caadb=0 and(bc)d=0.?

Observe that part(ii)of Corollary 2.2 is an extension of[13,Corollary 3.3]for complex matrices.

If the condition d=0 is added in Corollary 2.2,notice that parts(i)-(v)of Corollary 2.2 recover[6,Corollary 4.1-4.5],respectively,which include formulae for the generalized Drazin inverse of an anti-triangular operator matrix.

Now,we consider some expressions for the generalized Drazin inverse of triangular and anti-triangular matrices in Banach algebras which can be obtained using Theorem 2.1.

ProofIf b=0 in Theorem 2.1,we show this result.?

ProofThis result follows by Theorem 2.1 for c=0.? We can see that Corollary 2.3 and Corollary 2.4 are particular cases of[4,Theorem 2.3].

ProofUsing Theorem 2.1 for a=0,we obtain this corollary.?

We next develop necessary and sufcient conditions for the existence and the expressions for the group inverse of an anti-triangular matrix in Banach algebras.

ProofAssume that bc∈(pAp)#,d∈((1-p)A(1-p))#and dc=0.From Lemma 1.3, we deduce that cb∈((1-p)A(1-p))dand(cb)π(cb)2=c(bc)πbcb=0.Denote by u the right hand side of(2.3).Using Corollary 2.5,we have that x∈Adand

The assumption dc=0 implies that d#c=0 and

Observe that x∈A#if and only if x=xdx2.The equality x=xdx2is equivalent to

Since c=c(bc)#bc implies(cb)πcbdd#=c(bc)πbdd#=0,we conclude that x∈A#and x#=u if and only if c(bc)π=0 and(bc)πbdπ=0.?

The next corollary can be proved applying Theorem 2.1 for d=0.

In the following corollary,we obtain the same expression for the generalized Drazin inverse xdas in[14,Theorem 4.4]for the Drazin inverse of an operator matrix.

ProofThis result follows by Theorem 2.1.?

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∗Received June 2,2014;revised November 28,2014.The work was supported by the Ministry of Education and Science,Republic of Serbia(174007).