Geodesic
Cline's formula and Jacobson's lemma for g-Drazin inverse
Filomat, Tome 35 (2021) no. 15, p. 5083

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DOI

We present new conditions under which Cline's formula and Jacobson's lemma for g-Drazin inverse hold. Let A be a Banach algebra, and let a, b ∈ A satisfying a k b k a k = a k+1 for some k ∈ N. We prove that a has g-Drazin inverse if and only if b k a k has g-Drazin inverse. In this case, (b k a k) d = b k (a d) 2 a k and a d = a k [(b k a k) d ] k+1. Further, we study Jacobson's lemma for g-Drazin inverse in a Banach algebra under the preceding condition. The common spectral property of bounded linear operators on a Banach space is thereby obtained
DOI : 10.2298/FIL2115083C
Classification : 15A09, 47A10, 32A65, 16U99
Keywords: Cline’s formula, Jacobson’s leamm, g-Drazin inverse, Bounded linear operator, Common spectral property 
Huanyin Chen; Marjan Sheibani Abdolyousefi. Cline's formula and Jacobson's lemma for g-Drazin inverse. Filomat, Tome 35 (2021) no. 15, p. 5083 . doi: 10.2298/FIL2115083C
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     title = {Cline's formula and {Jacobson's} lemma for {g-Drazin} inverse},
     journal = {Filomat},
     pages = {5083 },
     year = {2021},
     volume = {35},
     number = {15},
     doi = {10.2298/FIL2115083C},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2115083C/}
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