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Additive Drazin inverse preservers. (English) Zbl 1131.47035

Let $H$ be a real or complex Hilbert space and denote by $B\left(H\right)$ the algebra of all bounded linear operators acting on $H$. An element $T\in B\left(H\right)$ is called Drazin invertible if there exists an element ${T}^{D}\in B\left(H\right)$ and a positive integer $k$ such that

$T{T}^{D}={T}^{D}T,\phantom{\rule{1.em}{0ex}}{T}^{D}T{T}^{D}={T}^{D},\phantom{\rule{1.em}{0ex}}{T}^{k+1}{T}^{D}={T}^{k}·$

The operator ${T}^{D}$ is unique and called the Drazin inverse of $T$. The author characterizes the additive maps $\phi :B\left(H\right)\to B\left(K\right)$ ($H,K$ being infinite-dimensional real or complex Hilbert spaces) which preserve the Drazin inverse in the sense that $\phi \left({T}^{D}\right)=\phi {\left(T\right)}^{D}$ holds for every Drazin invertible operator $T\in B\left(H\right)$. It is proved that if the range of $\phi$ contains every rank-one idempotent in $B\left(K\right)$ and $\phi$ does not annihilate all rank-one idempotents in $B\left(H\right)$, then $\phi$ is of one of the forms

$\phi \left(T\right)=\xi AT{A}^{-1},\phantom{\rule{1.em}{0ex}}A\in B\left(H\right),$
$\phi \left(T\right)=\xi A{T}^{tr}{A}^{-1},\phantom{\rule{1.em}{0ex}}A\in B\left(H\right),$

where $\xi =±1$ and $A:H\to K$ is a bounded linear or conjugate-linear bijection. The finite-dimensional case is also considered.

##### MSC:
 47B49 Transformers, preservers (operators on spaces of operators) 47A05 General theory of linear operators
##### Keywords:
additive preservers; Drazin inverse of operators