Throughout the paper, X is a Banach space and A denotes the Banach algebra L(X) of bounded linear operators on X. The commutant of a∈A is defined by comm(a)={x∈A​∣​xa=ax}. An element a in A has generalized Drazin inverse, i.e., g-Drazin inverse, if and only if there exists b∈comm(a) such that b=bab and a-a2b∈Aqnil. Here, Aqnil={a∈A​∣​1+ax∈U(A)​for every​x∈comm(a)}. For a Banach algebra A it is well known that a∈Aqnil⇔limn→∞∥an∥1n=0. The preceding b, if exists, is unique, and is denoted by ad. We call ad the g-Drazin inverse of a. We always use Ad to denote the set of all operators having g-Drazin inverses in A. It was proved that a∈Ad if and only if there exists an idempotent p∈comm(a) such that a+p is invertible and ap∈Aqnil, i.e., a∈A is quasipolar (see [15, Theorem 4.2]). Let a,b∈Ad. It is attractive to explore when the sum a+b has g-Drazin inverse. In [12, Theorem 2.3], Djordjević and Wei proved that if ab=0 then a+b∈Ad. In [11, Theorem 1], Deng and Wei proved that if ab=ba then a+b∈Ad if and only if 1+adb is g-Drazin invertible. In [25], Zou et al. proved that if a2b=aba and b2a=bab then a+b∈Ad. We refer the reader to [1, 4, 5, 7, 8, 10, 17, 18, 19] for further results.

In Section 2, we present some new additive results of g-Drazin inverses of the sum a+b under a number of polynomial conditions. These generalize the main results of Shakoor et al. (see [21, Lemma 5]).

In Section 3, we consider the g-Drazin inverse of a 2×2 operator matrix

where A∈L(X),D∈L(Y). Here, M is a bounded operator on X⊕Y. Such operator matrices have various applications in singular differential and difference equations, Markov chains, and iterative methods. The Drazin inverse of operator matrices has been well studied recently, e.g., [2, 3, 6, 9, 13, 22, 23, 24]. We generalize recent results of Deng et al. (see [10, Theorem 2, 3 and 5]), and of Yang and Liu (see [23, Theorem 3.3]).

If a∈A has g-Drazin inverse ad, then the element aπ=1-aad is called the spectral idempotent of a. In Section 4, we illustrate the g-Drazin inverse of a 2×2 operator matrix M under various conditions on spectral idempotents.

The aim of this section is to establish new additive results for g-Drazin inverses and give the explicit formulas for the g-Drazin inverse of the sum a+b. We begin with

Lemma 2.1. Let a,b∈A and ab=0. If a,b∈A have g-Drazin inverses, then a+b∈A has g-Drazin inverse and

Proof. See [12, Theorem 2.3].

Lemma 2.2. Let a,b,c∈A. If a,b∈A have g-Drazin inverses, then ac0b∈M2(A) has g-Drazin inverse.

Proof. See [12, Lemma 2.2].

Lemma 2.3. Let a,b∈Ad. If a2b=0,b2=0 and bab=0, then

Proof. Since (a+b)2=a2+(ab+ba),(ab+ba)2=0 and a2(ab+ba)=0,

As (ab+ba)2=0, then (ab+ba)d=0. By applying Lemma 2.1 and using (a2)d=(ad)2 we have,

Hence,

as required.

We are ready to prove:

Theorem 2.4. Let a,b∈Ad. If a3b=0,bab=0,ba2b=0, then a+b∈Ad and

where A=a20a+bb2,B=aba2b+ab20ab, and

Proof. Set

Then

Since a,b∈Ad, it follows by Lemma 2.2, that A has g-Drazin inverse. Clearly, B2=0, and so B has g-Drazin inverse. By a direct computation, we see that AB=00a2ba2b2, and so A2B=0 and BAB=0. Moreover, B2=0. Accordingly, M has g-Drazin inverse by Lemma 2.3.

Clearly, M=(a1(1,b))2. It follows by [14, Theorem 2.7] that a1(1,b) has g-Drazin inverse. By using Cline's formula, a+b=(1,b)a1 has g-Drazin inverse.

By virtue of [16, Theorem 2.1],

In light of Lemma 2.3, Md=Ad+B(Ad)2+AB(Ad)3. Moreover, we have

One easily checks that

Since (a,0)a1=a2∈Ad , it follows by Cline's formula, we see that

Likewise, we have

Clearly, HK=0. By virtue of [12, Theorem 2.3],

Corollary 2.5. Let a,b∈Ad. If a2b=0,b2a=0 and (ab)3=0, then a+b∈Ad.

Proof. Since (ab)3=0, we see that ab∈Ad. By using Cline's formula, ba∈Ad. Since a2b=0,b2a=0, it follows by Lemma 2.3, that a2+ab,ba+b2∈Ad. Let p=a2+ab and q=ba+b2. Then pq=ab3. Hence

and

In view of Theorem 2.4, (a+b)2=p+q∈Ad. This completes the proof by [14, Theorem 2.7].

Let a,b∈Ad. If a2b=0 and bab=0, then a+b∈Ad. This is a direct consequence of Theorem 2.4. Furthermore, we derive

Corollary 2.6. Let a,b∈Ad. If a2b=0,bab2=0 and (ab)3=0, then a+b∈Ad.

Proof. Let p=a2+ab and q=ba+b2. As in the proof in Corollary 2.5, we see that p,q∈Ad. We easily check that pq=ab3+ab2a, and so qpq=(ba+b2)(ab3+ab2a)=0, p2q=(a2+ab)(ab3+ab2a)=0. Therefore (a+b)2=p+q∈Ad. The proof is complete by Theorem 2.4.

Wang et al. studied the Drazin inverse of the sum of two bounded linear operators (see [22]). We now generalize the main results in [22] as follows.

Theorem 2.7. Let a,b∈Ad. If a3b=0,a2b+bab=0, then a+b∈Ad and

where A=a20a+bb2,B=aba2b+ab20ab, and

Proof. Set

Then

Since a3b=0,a2b+bab=0, we see that

Thus, we have AB=0. Since (ab)2=a(bab)=-a3b=0, we easily check that B3=0; hence, B has g-Drazin inverse. Since a,b∈A have g-Drazin inverses, it follows by Lemma 2.2, that A has g-Drazin inverse. In light of Lemma 2.3, M=A+B has g-Drazin inverse.

By using Lemma 2.3 again, Md=Ad+B(Ad)2+B2(Ad)3. Obviously, we have

As in the proof of Theorem 2.4, one easily checks that

Moreover,

as required.

Corollary 2.8. Let a,b∈Ad. If (ab)2=0,a2b+bab=0, then a+b∈Ad.

Proof. Clearly, a3b=-(ab)2=0, and therefore we obtain the result by Theorem 2.4.

We note that Corollary 2.8, is a nontrivial generalization of [12, Theorem 2.3], as the following shows.

Example 2.9. Let A and B be operators, acting on separable Hilbert space l2(ℕ), defined as follows respectively:

Then we easily check that (AB)2=0,A2B+BAB=0 and A,B∈L(l2(ℕ))d. Hence A+B has g-Drazin inverse by Corollary 2.8, in this case, AB≠0.

To illustrate the preceding results, we are concerned with the g-Drazin inverse for an operator matrix. Throughout this section, the operator matrix M is given by (1.1), i.e.,

where A∈L(X)d,D∈L(Y)d. Using different splitting approaches, we will obtain various conditions for the g-Drazin inverse of M.

Theorem 3.1. If A2BC=0,A2BD=0,CBC=0,CBD=0,CABC=0 and CABD=0. Then M has g-Drazin inverse.

Proof. Let p=AB00 and q=00CD, then M=p+q. By applying [10, Theorem 3], it is obvious that p,q have g-Dazin inverses. Now we have

and

Also

Then by Theorem 2.4, M has g-Drazine inverse.

Corollary 3.2. If ABC=0,ABD=0,CBC=0 and CBD=0. Then M has g-Drazin inverse.

Proof. This is obvious by Theorem 3.1.

Regarding a complex matrix as the operator matrix on ℂ×ℂ, we now show that corollary 3.2 is a non-trivial generalization of [10, Theorem 2]

Example 3.3. Let

be complex matrices and set

Then

In view of Corollary 3.2 M has g-Drazin inverse but BC≠0 and BD≠0.

Theorem 3.4. If A3B=0,CA2B=0,BCB=0,DCB=0, BCAB=0 and DCAB=0, then M has g-Drazin inverse.

Proof. Let p=A0C0 and q=0B0D, then M=p+q. In view of Lemma 2.2, p and q have g-Drazin inverses. Now we have

and

Also

Then by Theorem 2.4, M has g-Drazine inverse.

Corollary 3.5. If A2B=0,BCB=0,DCB=0 and CAB=0, then M has g-Drazin inverse.

Proof. It is a special case of Theorem 3.3.

Theorem 3.6. If (A2+BC)BD=0,CABD=0,DCBD=0, ABC=0 and CBC=0, then M has g-Drazin inverse.

Proof. Let p=ABC0 and q=000D, then M=p+q. Since ABC=0,CBC=0, it follows by Corollary 3.2 that p has g-Drazin inverse. By Lemma 2.2, q has g-Dazin inverse. Now we have

and

Also

Then by Theorem 2.4, M has g-Drazine inverse.

Corollary 3.7. If BC=0,BD=0, then M has g-Drazin inverse.

Proof. This is clear from Theorem 3.6.

We are now ready to prove:

Theorem 3.8. If A3B=0,CAB=0,CA2B=0,BCB=0 and DCB=0, then M has g-Drazin inverse.

Proof. Let p=A000 and q=0BCD, then M=p+q. Clearly, p has g-Drazin inverse. Since BCB=0 and DCB=0, it follows by Corollary 3.5 that q has g-Dazin inverse. We check that

and

Also

According to Theorem 2.4, M has g-Drazine inverse.

As an immediate consequence of Theorem 3.8, we now derive

Corollary 3.9. If AB=0 and CB=0, then M has g-Drazin inverse.

Let M be an operator matrix M given by (1.1). It is of interest to consider the g-Drazin inverse of M under generalized Schur condition D=CAdB (see [20]). The goal of this section is to consider another splitting of the block matrix M under such condition and present alternative theorems on spectral idempotents.

Theorem 4.1. Let A∈L(X),D∈L(Y) have g-Drazin inverses and M be given by (1.1). If I+AdBCAd is g-Drazin invertible,

then M has g-Drazin inverse.

Proof.

where

By assumption, we verify that P3Q=0,QPQ=0,QP2Q=0. Clearly, Q is quasinilpotent, and so it has g-Drazin inverse. Furthermore, we have

and P2P1=0,P22=0. Clearly, P2 has g-Drazin inverse. Moreover, we have

By hypothesis, we see that

It is obvious that, A2Ad=A(AAd) has g-Drazin inverse. Since (CAd) (AAdB)=CAdB=D has g-Drazin inverse, it follows by Cline's formula that AAdBCAd has g-Drazin inverse.

Since A2BCA=ABCA2, we easily check that

In light of [11, Theorem 1], A2Ad+AAdBCAd has g-Drazin inverse. By using Cline's formula again, P1 has g-Drazin inverse. Thus, by [12, Theorem 2.3], P g-Drazin inverse. By using Theorem 2.4, M has g-Drazin inverse, as asserted.

Corollary 4.2. Let A∈L(X),D∈L(Y) have g-Drazin inverses and M be given by (1.1). If I+AdBCAd is g-Drazin invertible,

then M has g-Drazin inverse.

Proof. Since BCA=0, we see that BCAπA=0 and A2BCA=ABCA2=0. Moreover, BCAπB=BC(I2-AAd)B=BCB=0. This completes the proof by Theorem 4.1.

Theorem 4.3. Let A∈L(X),D∈L(Y) have g-Drazin inverses and M be given by (1.1). If I+AdBCAd is g-Drazin invertible,

then M has g-Drazin inverse.

Proof. We easily see that

where

Then we check that P3Q=0,QPQ=0,QP2Q=0. Clearly, Q has g-Drazin inverse. Furthermore, we have

and P1P2=0. Clearly, P2 is nilpotent, and it has g-Drazin inverse. Obviously, we have

By hypothesis, we see that

As in the proof of Theorem 4.1, we easily check that A2Ad+AAdBCAd has g-Drazin inverse. Therefore P1 has g-Drazin inverse. By using Lemma 2.2, again, M has g-Drazin inverse, as asserted.

By using the other splitting approach of the block operator matrix, we now ready to prove:

Theorem 4.4. Let A∈L(X),D∈L(Y) have g-Drazin inverses and M be given by (1.1). If I+AdBCAd is g-Drazin invertible,

then M has g-Drazin inverse.

Proof. Let

where

Clearly, P is nilpotent, and so it has g-Drazin inverse.

Furthermore, we have

and Q1Q2=0. We easily see that Q1 is quasinilpotent, and it has g-Drazin inverse. Moreover,

By hypothesis, we see that

As in the proof of Theorem 4.1, we easily check that A2Ad+AAdBCAd has g-Drazin inverse. Therefore Q2 has g-Drazin inverse. By hypothesis, we check that P2=0 and QPQ=0, and so P3Q=0 and P2Q+QPQ=0. By virtue of Theorem 2.7, M has g-Drazin inverse, as asserted.

Analogously, we derive

Proposition 4.5. Let A∈L(X),D∈L(Y) have g-Drazin inverses and M be given by (1.1). If I+AdBCAd is g-Drazin invertible,

then M has g-Drazin inverse.

Proof. Let

where

As in the proof of Theorem 4.1, we easily check that P and Q have g-Drazin inverses. Since AdBCAπA=(Ad)2ABCAπA=0, we easily check that and P2=0 and QPQ=0. Therefore we complete the proof by Theorem 2.4.

Corollary 4.6. Let A∈L(X),D∈L(Y) have g-Drazin inverses and M be given by (1.1). If I+AdBCAd is g-Drazin invertible,

then M has g-Drazin inverse.

Proof. This is obvious by Proposition 4.5.