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Zbl 1131.47035
Cui, Jianlian
(English)
[J] Linear Algebra Appl. 426, No. 2-3, 448-453 (2007). ISSN 0024-3795

Let $H$ be a real or complex Hilbert space and denote by $B(H)$ the algebra of all bounded linear operators acting on $H$. An element $T\in B(H)$ is called Drazin invertible if there exists an element $T^D\in B(H)$ and a positive integer $k$ such that $$TT^D=T^DT, \quad T^DTT^D=T^D, \quad T^{k+1}T^D=T^k.$$ The operator $T^D$ is unique and called the Drazin inverse of $T$. The author characterizes the additive maps $\phi:B(H)\to B(K)$ ($H,K$ being infinite-dimensional real or complex Hilbert spaces) which preserve the Drazin inverse in the sense that $\phi(T^D)=\phi(T)^D$ holds for every Drazin invertible operator $T\in B(H)$. It is proved that if the range of $\phi$ contains every rank-one idempotent in $B(K)$ and $\phi$ does not annihilate all rank-one idempotents in $B(H)$, then $\phi$ is of one of the forms $$\phi(T)=\xi ATA^{-1}, \quad A\in B(H),$$ $$\phi(T)=\xi AT^{tr}A^{-1}, \quad A\in B(H),$$ where $\xi=\pm 1$ and $A:H\to K$ is a bounded linear or conjugate-linear bijection. The finite-dimensional case is also considered.
[Lajos Molnár (Debrecen)]
MSC 2000:
*47B49 Transformers (=operators on spaces of operators)
47A05 General theory of linear operators

Keywords: additive preservers; Drazin inverse of operators

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