A note on Cline’s formula for the generalized Drazin inverse
A note on Cline’s formula for the generalized Drazin inverse
复制标题
DOI：
10.1080/03081087.2014.922965
复制
发表时间：
2015-06
影响因子：
1.1
通讯作者：
D. Mosić
中科院分区：
文献类型：
--
作者：
D. Mosić
Liao等人证明了如果乘积是广义Drazin可逆的，那么将Cline公式推广到广义Drazin可逆的情形也是如此。本文证明了在Banach代数中，如果广义Drazin可逆，则广义Drazin可逆。因此，我们将Cline公式推广到元素和乘积的幂的广义Drazin可逆性的情形。证明了广义Drazin可逆的充要条件是广义Drazin可逆;广义Drazin可逆的充要条件是广义Drazin可逆。
Liao et al. proved that if the product is generalized Drazin invertible, then so is extending the Cline’s formula to the case of the generalized Drazin invertibility. In this paper, we show that if is generalized Drazin invertible for arbitrary , then is generalized Drazin invertible in a Banach algebra. So, we generalize Cline’s formula to the case of the generalized Drazin invertibility of the powers of the products of elements and . Also, we prove that is generalized Drazin invertible if and only if is generalized Drazin invertible; and is generalized Drazin invertible if and only if is generalized Drazin invertible.