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Nonlinear maps preserving similarity on $\mathcal B(H)$. (English) Zbl 1124.47025
Let $X$ be a Banach space of dimension $\ge 3$ and let $\sim$ denote the similarity relation. The authors classify surjections $\Phi:B(X)\to B(X)$ which enjoy the property $T+S\sim R$ (respectively, $T-S\sim R$) precisely when the same holds for their $\Phi$-images. It is interesting that no linearity on the mapping $\Phi$ is imposed in advance. For infinite-dimensional spaces, it turns out that such maps are scalar multiples of linear or conjugate-linear Jordan isomorphisms. When $\dim X<\infty$, one must supplement the result by adding an additive trace-like function. The proof is a reduction to bijective maps which preserve rank-one nilpotents in the following way: given two of them, $T$ and $S$, one has that $T+S$ is a rank-one nilpotent precisely when the same holds for their images.
MSC:
47B49Transformers, preservers (operators on spaces of operators)
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References:
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