Abstract

We investigate additive properties of the generalized Drazin inverse in a Banach algebra . We find explicit expressions for the generalized Drazin inverse of the sum , under new conditions on . As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

1. Introduction

Let be a complex Banach algebra with unite . We use to denote the spectrum of . The sets of all nilpotent and quasinilpotent elements of will be denoted by and , respectively.

The generalized Drazin inverse of (introduced by Koliha in [1]) is the element which satisfies If there exists the generalized Drazin inverse, then the generalized Drazin inverse of is unique and is denoted by . The set of all generalized Drazin invertible elements of is denoted by . For interesting properties of the generalized Drazin inverse see [2–6]. For a complete treatment of the generalized Drazin inverse, see [7, Chapter 2].

If is an idempotent, we denote . We can represent element as where , , , and .

Let and be the spectral idempotent of corresponding to . It is well known that can be represented in the following matrix form ([7, Chapter 2]): relative to , where is invertible in the algebra , is its inverse in , and is quasinilpotent in the algebra . Thus, the generalized Drazin inverse of can be expressed as Obviously, if , then is generalized Drazin invertible and .

In this paper, we first give the formulas of under the conditions and , respectively. Then we will apply these formulas to provide some representations for the generalized Drazin inverse of the operator matrix under some conditions.

2. Main Results

First we start the following result which is proved in [8] for matrices, extended in [9] for a bounded linear operator and in [10] for arbitrary elements in a Banach algebra.

Lemma 1 (see [10, Theorem 2.3]). Let and be an idempotent. Assume that and are represented as (i)If and , then and are generalized Drazin invertible, and where (ii)If and , then and and are given by (6) and (7).

Lemma 2 (see [11, Lemma 2.1]). Let . If or , then .

The following result is a generalization of [10, Corollary 3.4].

Theorem 3. If , , and , then and

Proof. First, suppose that . Therefore, and from we obtain . Using Lemma 2, and (8) holds.
Now we assume is not quasinilpotent, using matrix representations of and relative to . We have where , .
Let us represent From and we obtain and . Since is invertible, we have and .
Hence we have
The condition implies that . Hence, using Lemma 2, we get . By Lemma 1, we obtain that and where Now from (14), using the matrix representation of , , and , we easily obtain formula (8) of the theorem.

The next result is a generalization of [12, Theorem 2.2] and [10, Example 4.5].

Theorem 4. Let . If , then and

Proof. If is quasinilpotent, we can apply Theorem 3 and we obtain (15) for this particular case. Now we assume that is neither invertible nor quasinilpotent and consider the following matrix representations of , , and relative to the :
The condition implies that and . Since is invertible, we have and .
Thus, can be represented as Therefore, and, from Lemma 1, we have
From and we obtained . From Theorem 3, we get and Further, applying Lemma 1 to , we get where Observe that since , then .
Hence, the expression of reduces to From we get .
Hence, from formula (20) and , we have Then substituting (20) and (24) in (22), we get
Now, replacing by the above expression and considering matrix representations of and , after direct computations, we obtain the formula (15) for .

3. Applications

In this section, we give some formulas for the generalized Drazin inverse of a operator matrix under some conditions.

Finding an explicit representation for the generalized Drazin inverse of an operator matrix in terms of , , , and related generalized Drazin inverse has been studied by several authors [9, 13–15]. Djordjević and Stanimirović [9] generalize the well-known result in [8, 16] concerning the Drazin inverse of block upper triangular matrices to the generalized Drazin inverse for a block triangular operator matrix. Further, they consider the case where , , and .

This section is devoted to the generalized Drazin inverse of operator matrix: where and are generalized Drazin invertible.

Next we will state some auxiliary lemmas.

Lemma 5 (see [2, 3]). Let and be generalized Drazin invertible and let be matrix of form (26). If and , then where

Lemma 6 (see [17, Lemma 3.1]). If is matrix of form (26), such that is generalized Drazin invertible, is quasinilpotent, and for any nonnegative integer , then is generalized Drazin invertible and where

Lemma 7. Let . Then , , , and and .

Proof. The Jordan canonical form of permits us to write , where and are nonsingular, and is nilpotent with index . Thus . Now, it is evident that and , which lead to the affirmations of this lemma.

In [9, Theorem 5.3] authors gave an explicit representation for under conditions , , and . Here we replace the last two conditions by the two weaker conditions and .

Theorem 8. Let and be generalized Drazin invertible and let be matrix of form (26). If , and . Then

Proof. We can split matrix as , where
Since and , we have
From and applying Lemma 5 to , we obtain where is defined in (28). From and , we get .
Since and , we obtain . Applying Theorem 4, we get
From , we have Hence from (35), we obtain Since , we have
The conditions and   imply that . From and , we get
From (36), (38), and (39) it follows (31).
The proof is finished.

Since we can obtain the following result, applying Theorem 8 to .

Theorem 9. Let and be generalized Drazin invertible and let be matrix of form (26). If , and . Then

Theorem 10. Let , , and be generalized Drazin invertible and let be matrix of form (26). If , and . Then

Proof. We can split matrix as , where Since from , it is easy to get . Since is nilpotent, we have . Applying Theorem 4 to the particular case, we get
The conditions and   imply that , for , so we get
From (44) and (47) it follows (42).
The proof is finished.

Theorem 11. Let and be generalized Drazin invertible and let be matrix of form (26). If , and for any nonnegative integer . Then where and are defined in (30).

Proof. We can split matrix as , where
From and , we have so we get .
Note that is quasinilpotent, , and for any nonnegative integer ; we can apply Lemma 6 to with replaced by ; we have where Observe that (50) and (51) yield so we get The condition implies that Hence we have
From and , we obtain . Applying Theorem 4, we get where so we get Hence from (58) and (60) we obtain
By direct computation we verify that From , we have Observe that (51) and yield From (62) and (64) it follows (48).
The proof is finished.