Abstract
Let \({\mathscr {L}}({\mathscr {H}})\) be the algebra of all bounded linear operators on a complex Hilbert space \({\mathscr {H}}\) with \(\dim {\mathscr {H}}\ge 3\), and let \(\mathscr {A} \) and \(\mathscr {B}\) be two subsets of \({\mathscr {L}}({\mathscr {H}})\) containing all operators of rank at most one. For \(\varepsilon \in (0,1)\) the \(\varepsilon \)-condition spectrum of any \(A\in {\mathscr {L}}({\mathscr {H}})\) is defined by
where \(\sigma (A)\) is the spectrum of A. The \(\varepsilon \)-condition spectral radius of A is given by
We compute the \(\varepsilon \)-condition spectrum of any operator of rank at most one, and give an explicit formula for its \(\varepsilon \)-condition spectral radius. It is then illustrated that the results can be applied to characterize surjective mappings \(\phi :\mathscr {A} \longrightarrow \mathscr {B}\) satisfying
where \(\delta \) stands for \(\sigma _\varepsilon (\cdot )\) or \(r_\varepsilon (\cdot ).\)
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Communicated by Daniel Aron Alpay.
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Abdelali, Z.E.A., Nkhaylia, H. Condition Spectrum of Rank One Operators and Preservers of the Condition Spectrum of Skew Product of Operators. Complex Anal. Oper. Theory 14, 69 (2020). https://doi.org/10.1007/s11785-020-01028-9
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DOI: https://doi.org/10.1007/s11785-020-01028-9