Drazin invertibility of sum and product of closed linear operators

Bekkai MESSIRDIa,*, Sanaa MESSIRDIb and Miloud MESSIRDIc

Drazin invertibility of sum and product of closed linear operators

Authors :

Bekkai MESSIRDI^a,*, Sanaa MESSIRDI^b and Miloud MESSIRDI^c

Author Address :

^aD´epartement de Math´ematiques, Universit´e d’Oran (Es-s´enia), B.P 1524, El Menouar, Oran. Oran 31000, Algeria.
^b,cD´epartement de Math´ematiques, Universit´e de Tlemcen . Tlemcen 13000, Algeria.

*Corresponding author.

Abstract :

The paper present a survey of results concerning the fundamental properties of the Drazin inverse for bounded operators and an intersting studies of the Drazin inverse for a closed operator in a Banach space. Some necessary and sufficient conditions for }$A${small closed linear operator to possess a Drazin inverse }$A^{D}${small are given, we obtain also a useful caracterization and explicit formula for the Drazin inverse }$ (A+B)^{D}$ {small and } $(AB)^{D}$ {small if }$A${small and }$B$ {small are closed operators.