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Condition Spectrum of Rank One Operators and Preservers of the Condition Spectrum of Skew Product of Operators

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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

$$\begin{aligned} \sigma _{\epsilon }(A) := \sigma (A)\cup \left\{ \lambda \in \mathbb {C}\setminus \sigma (A):~\Vert (\lambda I -A)^{-1}\Vert \Vert \lambda I -A\Vert \ge \frac{1}{\varepsilon }\right\} , \end{aligned}$$

where \(\sigma (A)\) is the spectrum of A. The \(\varepsilon \)-condition spectral radius of A is given by

$$\begin{aligned} r_\varepsilon (A):=\sup \left\{ |z| : z\in \sigma _\varepsilon (A) \right\} . \end{aligned}$$

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

$$\begin{aligned} \delta (\phi (A)^*\phi (B)) = \delta (A^*B) \quad \text{ for } \text{ all } A,B\in \mathscr {A} \end{aligned}$$

where \(\delta \) stands for \(\sigma _\varepsilon (\cdot )\) or \(r_\varepsilon (\cdot ).\)

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Correspondence to Zine El Abidine Abdelali.

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Communicated by Daniel Aron Alpay.

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“Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.

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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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