Abstract
Let F be a field and n ⩾ 3 . Suppose S 1 , S 2 ⊆ M n ( F ) contain all rank-one idempotents. The structure of surjections ϕ : S 1 → S 2 satisfying ABA = 0 ⇔ ϕ ( A ) ϕ ( B ) ϕ ( A ) = 0 is determined. Similar results are also obtained for (a) subsets of bounded operators acting on a complex or real Banach space X , (b) the space of Hermitian matrices acting on n-dimensional vectors over a skew-field D , (c) subsets of self-adjoint bounded linear operators acting on an infinite dimensional complex Hilbert space. It is then illustrated that the results can be applied to characterize mappings ϕ on matrices or operators such that F ( ABA ) = F ( ϕ ( A ) ϕ ( B ) ϕ ( A ) ) for all A , B for functions F such as the spectral norm, Schatten p-norm, numerical radius and numerical range, etc.
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