Maps preserving the \(\partial\)-spectrum of product or triple product of operators. (English) Zbl 07751543
Let \({\mathcal B}(X)\) be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space \(X\). For any \(T\in{\mathcal B}(X)\), let \(\sigma(T)\) be its spectrum and \(\partial\sigma(T)\) be the boundary of its spectrum. A map \(\Delta\) from \({\mathcal B}(X)\) into the collection of all closed subsets of \(\mathbb{C}\) is said to be \(\partial\)-spectrum if \[\partial\sigma(T)\subseteq\Delta(T)\subseteq\sigma(T)\] for all \(T\in{\mathcal B}(X)\). In the paper under review, the authors characterize all surjective maps \(\phi\) on \({\mathcal B}(X)\) that satisfy \[\Delta(\phi(T)\phi(S))=\Delta(TS)\] for all \(S,~T\in{\mathcal B}(X)\). They also obtain a similar result but when the product \(TS\) is replaced by the triple product \(TST\).
Reviewer: Abdellatif Bourhim (Syracuse)
MSC:
| 47B49 | Transformers, preservers (linear operators on spaces of linear operators) |
| 47B48 | Linear operators on Banach algebras |
| 46H05 | General theory of topological algebras |
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