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Additive Drazin inverse preservers. (English) Zbl 1131.47035
Let $H$ be a real or complex Hilbert space and denote by $B(H)$ 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.

47B49Transformers, preservers (operators on spaces of operators)
47A05General theory of linear operators
Full Text: DOI
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