generalized drazin invertible operators
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2020 ◽  
Vol 54 (1) ◽  
pp. 98-106
Author(s):  
So. Messirdi ◽  
Sa. Messirdi ◽  
B. Messirdi

In this paper we present some new characteristics and expressions of left and right generalized Drazin invertible bounded operators on a Banach space $X.$ An explicit formula relating the left and the right generalized Drazin inverses to spectral idempotents is provided. In addition, we give a characterization of operators in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) with equal spectral idempotents, where $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) denotes the set of all left (resp. right) generalized Drazin invertible bounded operators on $X.$ Next, we give some sufficient conditions which ensure that the product of elements of $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) remains in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$). Finally, we extend Jacobson's lemma for left and right generalized Drazin invertibility. The provided results extend certain earlier works given in the literature.


2018 ◽  
Vol 13 (3) ◽  
pp. 1361-1375 ◽  
Author(s):  
Djalel Ounadjela ◽  
Mohammed Benharrat ◽  
Bekkai Messirdi

2017 ◽  
Vol 67 (1) ◽  
pp. 159-172 ◽  
Author(s):  
Djalel Ounadjela ◽  
Kouider Miloud Hocine ◽  
Bekkai Messirdi

2014 ◽  
Vol 63 (8) ◽  
pp. 1635-1648 ◽  
Author(s):  
Kouider Miloud Hocine ◽  
Mohammed Benharrat ◽  
Bekkai Messirdi

2006 ◽  
Vol 80 (3) ◽  
pp. 383-396 ◽  
Author(s):  
N. Castro-González ◽  
J. Y. Vélez-Cerrada

AbstractLet aπ denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We characterize elements b such that 1 − (bπ − aπ)2 is invertible. We also apply this result in rings with involution to obtain a characterization of the perturbation of EP elements. In Banach algebras we obtain a characterization in terms of matrix representations and derive error bounds for the perturbation of the Drazin Inverse. This work extends recent results for matrices given by the same authors to the setting of rings and Banach algebras. Finally, we characterize generalized Drazin invertible operators A, B ∈ (X) such that pr(Bπ) = pr(Aπ + S), where pr is the natural homomorphism of (X) onto the Calkin algebra and S ∈(X) is given.2000 Mathematics subject classification: primary 16A32, 16A28, 15A09.


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