Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized Cline's formula for G-Drazin inverse in a ring   In this paper, we give a generalized Cline's formula for the generalized
Drazin inverse. Let $R$ be a ring, and let $a,b,c,d\in R$ satisfying
$$\begin{array}{c} (ac)^2 = (db)(ac), (db)^2 = (ac)(db);\\ b(ac)a = b(db)a,
c(ac)d = c(db)d.\end{array}$$ Then $ac\in R^d$ if and only if $bd\in R^d$. In
this case, $(bd)^d = b((ac)^d)^2 d$. We also present generalized Cline's
formulas for Drazin and group inverses. Some weaker conditions in a Banach
algebra are also investigated. These extend the main results of Cline's formula
on g-Drazin inverse of Liao, Chen and Cui (Bull. Malays. Math. Soc., 37 (2014),
37-42), Lian and Zeng (Turk. J. Math., 40 (2016), 161-165) and Miller and
Zguitti (Rend. Circ. Mat. Palermo, II. Ser., 67 (2018), 105-114). As an
application, new common spectral property of bounded linear operators over
Banach spaces are obtained.

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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an