### Article

Kyungpook Mathematical Journal 2024; 64(2): 205-218

**Published online** June 30, 2024 https://doi.org/10.5666/KMJ.2024.64.2.205

Copyright © Kyungpook Mathematical Journal.

### The G-Drazin Inverse of an Operator Matrix over Banach Spaces

Farzaneh Tayebi, Nahid Ashrafi and Rahman Bahmani, Marjan Sheibani Abdolyousefi*

Department of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran

e-mail : ftayebis@gmail.com, nashrafi@semnan.ac.ir and rbahmani@semnan.ac.ir

Farzanegan Campmus, Semnan University, Semnan, Iran

e-mail : m.sheibani@semnan.ac.ir

**Received**: June 29, 2023; **Revised**: January 22, 2024; **Accepted**: January 22, 2024

### Abstract

Let

**Keywords**: generalized Drazin inverse, additive property, operator matrix, spectral idempotent

### 1. Introduction

Throughout the paper, *X* is a Banach space and *X*. The commutant of *a* in *b*=*bab* and *b*, if exists, is unique, and is denoted by *a*. We always use *a*+*p* is invertible and *a*+*b* has g-Drazin inverse. In [12, Theorem 2.3], Djordjević and Wei proved that if *ab*=0 then *ab*=*ba* then

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

where *M* is a bounded operator on

If *a*. In Section 4, we illustrate the g-Drazin inverse of a *M* under various conditions on spectral idempotents.

### 2. G-Drazin inverses

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 *

*Proof.* See [12, Theorem 2.3].

**Lemma 2.2.** *Let *

*Proof.* See [12, Lemma 2.2].

**Lemma 2.3.** *Let *

*bab*=0, then

*Proof.* Since

As

Hence,

as required.

We are ready to prove:

**Theorem 2.4.** *Let *

*where *

*Proof.* Set

Then

Since *A* has g-Drazin inverse. Clearly, *B* has g-Drazin inverse. By a direct computation, we see that *BAB*=0. Moreover, *M* has g-Drazin inverse by Lemma 2.3.

Clearly,

By virtue of [16, Theorem 2.1],

In light of Lemma 2.3,

One easily checks that

Since

Likewise, we have

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

**Corollary 2.5.** *Let *

*Proof.* Since

and

In view of Theorem 2.4,

Let *bab*=0, then

**Corollary 2.6.** *Let *

*Proof.* Let

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 *

*where *

*Proof.* Set

Then

Since

Thus, we have *AB*=0. Since *B* has g-Drazin inverse. Since *A* has g-Drazin inverse. In light of Lemma 2.3, *M*=*A*+*B* has g-Drazin inverse.

By using Lemma 2.3 again,

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

Moreover,

as required.

**Corollary 2.8.** * Let *

*Proof.* Clearly,

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 *

*Then we easily check that *

*A*+

*B*has g-Drazin inverse by Corollary 2.8, in this case,

### 3. Splitting Approach

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 *M*.

**Theorem 3.1.** *If *

*Proof.* Let *M*=*p*+*q*. By applying [10, Theorem 3], it is obvious that

and

Also

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

**Corollary 3.2.** * If *

*Proof.* This is obvious by Theorem 3.1.

Regarding a complex matrix as the operator matrix on

**Example 3.3.** *Let*

*be complex matrices and set*

Then

*In view of Corollary 3.2 M has g-Drazin inverse but *

**Theorem 3.4.** * If *

*Proof.* Let *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 *

*Proof.* It is a special case of Theorem 3.3.

**Theorem 3.6.** * If *

*Proof.* Let *M*=*p*+*q*. Since *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 *

*Proof.* This is clear from Theorem 3.6.

We are now ready to prove:

**Theorem 3.8.** * If *

*Proof.* Let *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.*

### 4. Spectral Conditions

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 *M* under such condition and present alternative theorems on spectral idempotents.

**Theorem 4.1.** *Let *

*then M has g-Drazin inverse.*

*Proof.*

where

By assumption, we verify that *Q* is quasinilpotent, and so it has g-Drazin inverse. Furthermore, we have

and

By hypothesis, we see that

It is obvious that,

Since

In light of [11, Theorem 1], *P* g-Drazin inverse. By using Theorem 2.4, *M* has g-Drazin inverse, as asserted.

**Corollary 4.2.** *Let *

*then M has g-Drazin inverse.*

*Proof.* Since *BCA*=0, we see that

**Theorem 4.3.** *Let *

*then M has g-Drazin inverse.*

*Proof.* We easily see that

where

Then we check that *Q* has g-Drazin inverse. Furthermore, we have

and

By hypothesis, we see that

As in the proof of Theorem 4.1, we easily check that *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.** *M* be given by (1.1). If

*then M has g-Drazin inverse.*

*Proof.* Let

where

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

Furthermore, we have

and

By hypothesis, we see that

As in the proof of Theorem 4.1, we easily check that *QPQ*=0, and so *M* has g-Drazin inverse, as asserted.

Analogously, we derive

**Proposition 4.5.** *M* be given by (1.1). If

*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 *QPQ*=0. Therefore we complete the proof by Theorem 2.4.

**Corollary 4.6.** *M* be given by (1.1). If

*then M has g-Drazin inverse.*

*Proof.* This is obvious by Proposition 4.5.

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