Further results on Left and Right Generalized Drazin Invertible Operators

So. Messirdi; Sa. Messirdi; B. Messirdi

doi:10.30970/ms.54.1.98-106

So. Messirdi Department of Mathematics and Informatics, University of Mostaganem
Sa. Messirdi High Industrial Institute for Social Promotion ISIPS-Hainaut
B. Messirdi High School of Electrical and Energetic Engineering-Oran

DOI: https://doi.org/10.30970/ms.54.1.98-106

Keywords: left generalized Drazin inverse; right generalized Drazin inverse; spectral idempotent; product, Jacobson’s lemma

Abstract

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.

Author Biographies

So. Messirdi, Department of Mathematics and Informatics, University of Mostaganem

Department of Mathematics and Informatics, University of Mostaganem. Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO)

Sa. Messirdi, High Industrial Institute for Social Promotion ISIPS-Hainaut

High Industrial Institute for Social Promotion ISIPS-Hainaut

B. Messirdi, High School of Electrical and Energetic Engineering-Oran

High School of Electrical and Energetic Engineering-Oran
Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO)
University of Oran1, Ahmed Benbella, Algeria

References

P. Aiena, Fredholm and Local Spectral Theory II: With Application to Weyl-type Theorems, Lecture Notes in Math. 2235, Springer, 2018. ISBN 978-3-030-02266-2.

M. Benharrat, K. Miloud Hocine, B. Messirdi, Left and right generalized Drazin invertible operators and local spectral theory, Proyecciones J. Math., 38 (2019), 897-919. DOI: 10.22199/issn.0717-6279-2019-05-0058.

G. Corach, B. Duggal, R. Harte, Extensions of Jacobson's Lemma, Commun. Algebra, 41 (2013), №2, 520-531. DOI: 10.1080/00927872.2011.602274.

D.S. Cvetkovic-Ilic, R.E. Harte, On Jacobson's lemma and Drazin invertibility, Appl. Math. Lett., 23 (2010), 417-420. DOI: 10.1016/j.aml.2009.11.009.

D. Drivaliaris and N. Yannakakis, Subspaces with a Common Complement in a Banach Space, Studia Math., 182 (2007), 141-164. DOI: 10.4064/sm182-2-4.

J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J., 38 (1996), 367-381. DOI: 10.1017/S0017089500031803.

K. Miloud Hocine, M. Benharrat, B. Messirdi, Left and right generalized Drazin invertible operators, Linear Multilinear A., 63 (2015), 1635-1648. DOI: 10.1080/03081087.2014.962534.

D. Ounadjela, M. Benharrat, B. Messirdi, The perturbation classes problem for generalized Drazin invertible operators II, Complex Anal. Oper. Th., 13 (2019), 1361-1375. DOI: 10.1007/s11785-018-0879-1.

D. Ounadjela, K. Miloud Hocine, B. Messirdi, The perturbation classes problem for generalized Drazin invertible operators I, Rend. Circ. Mat. Palermo, II. Ser., 67 (2018), 159-172. DOI 10.1007/s12215-017-0302-1

G. Zhuang, J. Chen, J. Cui, Jacobson's lemma for the generalized Drazin inverse, Linear Algebra Appl. 436 (3) (2012), 742-746. DOI: 10.1016/j.laa.2011.07.044.