Maps preserving the spectrum of skew Lie product of operators. (English) Zbl 1513.47074
Summary: Let \(\mathcal{B}(\mathcal{H})\) denote the algebra of all bounded linear operators acting on a complex Hilbert space \(\mathcal{H}\). In this paper, we show that a surjective map \(\varphi\) on \(\mathcal{B}(\mathcal{H})\) satisfies
\[
\sigma(\varphi(T)\varphi(S)-\varphi(S)\varphi(T)^\ast) = \sigma(TS-ST^\ast), \quad T,S \in \mathcal{B}(\mathcal{H}),
\] if and only if there exists a unitary operator \(U\in \mathcal{B}(\mathcal{H})\) such that
\[
\varphi(T)=\lambda UTU^\ast, \quad T\in \mathcal{B}(\mathcal{H}),
\] where \(\lambda\in\left\{-1, 1\right\}\).
MSC:
| 47B49 | Transformers, preservers (linear operators on spaces of linear operators) |
| 47A10 | Spectrum, resolvent |
| 47A11 | Local spectral properties of linear operators |
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