Geodesic
Generalized Hirano inverses in Banach algebras
Filomat, Tome 33 (2019) no. 19, p. 6239

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DOI

Let A be a Banach algebra. An element a ∈ A has generalized Hirano inverse if there exists b ∈ A such that b = bab, ab = ba, a2 − ab ∈ Aqnil. We prove that a ∈ A has generalized Hirano inverse if and only if a − a3 ∈ Aqnil, if and only if a is the sum of a tripotent and a quasinilpotent that commute. The Cline’s formula for generalized Hirano inverses is thereby obtained. Let a, b ∈ A have generalized Hirano inverses. If a2b = aba and b2a = bab, we prove that a + b has generalized Hirano inverse if and only if 1 + adb has generalized Hirano inverse. The generalized Hirano inverses of operator matrices on Banach spaces are also studied
DOI : 10.2298/FIL1919239C
Classification : 15A09, 32A65, 16E50
Keywords: generalized Drazin inverse, tripotent, Cline’s formula, additive property, operator matrix 
Huanyin Chen; Marjan Sheibani. Generalized Hirano inverses in Banach algebras. Filomat, Tome 33 (2019) no. 19, p. 6239 . doi: 10.2298/FIL1919239C
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     title = {Generalized {Hirano} inverses in {Banach} algebras},
     journal = {Filomat},
     pages = {6239 },
     year = {2019},
     volume = {33},
     number = {19},
     doi = {10.2298/FIL1919239C},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL1919239C/}
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