Abstract: We present in this talk additive properties for the
$g$-Drazin
inverse in a complex Banach algebra. Some recent papers
deal with the
problem of finding an explicit expression for the
$g$-Drazin inverse of
$a+b$ in terms of $a$, $b$ $a^D$, and $b^D$. We
will provide
representations of $(a+b)^D$ under conditions $a^2b
=ab^2=0$, and when
the weaker conditions $a^Db=0$, $a^2
ba^{\pi}=ab^2a^{\pi}=0$ are
assumed. The auxiliary results used in
our development involve the
resolvent of a $2\times 2$ matrix with
entries in a Banach algebra, and
the square of its $g$-Drazin
inverse. Our results recover the case
$ab=0$ studied by Hartwig et
al. We will comment on the application of
the additive results to
obtain representations of the Drazin inverse of
a $2\times2$ complex
block matrix in terms of the individual blocks. |