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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 TB(H) is called Drazin invertible if there exists an element T D B(H) and a positive integer k such that

TT D =T D T,T D TT D =T D ,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 φ:B(H)B(K) (H,K being infinite-dimensional real or complex Hilbert spaces) which preserve the Drazin inverse in the sense that φ(T D )=φ(T) D holds for every Drazin invertible operator TB(H). It is proved that if the range of φ contains every rank-one idempotent in B(K) and φ does not annihilate all rank-one idempotents in B(H), then φ is of one of the forms

φ(T)=ξATA -1 ,AB(H),
φ(T)=ξAT tr A -1 ,AB(H),

where ξ=±1 and A:HK is a bounded linear or conjugate-linear bijection. The finite-dimensional case is also considered.

MSC:
47B49Transformers, preservers (operators on spaces of operators)
47A05General theory of linear operators