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Additive mappings preserving rank-one idempotents. (English) Zbl 1125.47028

This paper presents a remarkable contribution to the study of additive preservers. It is well-known that the different kinds of rank-one preservers play important roles in the solution of many preserver problems. In the paper under review, the author deals with additive maps preserving idempotents of rank at most one. Let \(X,Y\) be Banach spaces. Denote by \(B(Y)\) the space of all bounded linear operators on \(Y\) and by \(F(X)\) the space of all bounded linear operators on \(X\) of finite rank. In the case when \(X\) is infinite-dimensional, the author gives the complete description of all additive transformations \(\phi:F(X)\to B(Y)\) which map rank-one idempotents to idempotents of rank at most one. It is remarked that, with some natural modifications, the result remains true also in the case when \(3\leq \dim X<\infty\).

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
46B28 Spaces of operators; tensor products; approximation properties
15A03 Vector spaces, linear dependence, rank, lineability
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