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New characterizations of g-Drazin inverse in a Banach algebra
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In this paper, we present a new characterization of g-Drazin inverse in a
Banach algebra. We prove that an element a in a Banach algebra has g-Drazin
inverse if and only if there exists x ? A such that ax = xa, a-a2x ? A
qnil. As an application, we obtain the sufficient and necessary conditions
for the existence of the g-Drazin inverse for certain 2 x 2 anti-triangular
matrices over a Banach algebra. These extend the results of Koliha (Glasgow
Math. J., 38(1996), 367-381), Nicholson (Comm. Algebra, 27(1999), 3583-3592
and Zou et al. (Studia Scient. Math. Hungar., 54(2017), 489-508).
Title: New characterizations of g-Drazin inverse in a Banach algebra
Description:
In this paper, we present a new characterization of g-Drazin inverse in a
Banach algebra.
We prove that an element a in a Banach algebra has g-Drazin
inverse if and only if there exists x ? A such that ax = xa, a-a2x ? A
qnil.
As an application, we obtain the sufficient and necessary conditions
for the existence of the g-Drazin inverse for certain 2 x 2 anti-triangular
matrices over a Banach algebra.
These extend the results of Koliha (Glasgow
Math.
J.
, 38(1996), 367-381), Nicholson (Comm.
Algebra, 27(1999), 3583-3592
and Zou et al.
(Studia Scient.
Math.
Hungar.
, 54(2017), 489-508).
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