Linear maps preserving the minimum and surjectivity moduli of Hilbert space operators. (English) Zbl 1165.47025
Summary: Let \(\mathcal B(H)\) be the algebra of all bounded linear operators on a complex infinite-dimensional Hilbert space \(H\). For every \(T\in \mathcal B(H)\), let \(m(T)\) and \(q(T)\) denote the minimum modulus and surjectivity modulus of \(T\), respectively. Let \(\phi:\mathcal B(H)\to \mathcal B(H)\) be a surjective linear map. In this paper, we prove that the following assertions are equivalent:
(i) \(m(T)\) = \(m(\phi(T))\) for all \(T\in\mathcal B(H)\),
(ii) \(q(T)\) = \(q(\phi(T))\) for all \(T\in\mathcal B(H)\),
(iii) there exist two unitary operators \(U,V\in\mathcal B(H)\) such that \(\phi(T)=UTV\) for all \(T\in\mathcal B(H)\).
This generalizes the result of M.Mbekhta [“Linear maps preserving the minimum and surjectivity moduli of operators” (preprint), Theorem 3.1] to the non-unital case.
(i) \(m(T)\) = \(m(\phi(T))\) for all \(T\in\mathcal B(H)\),
(ii) \(q(T)\) = \(q(\phi(T))\) for all \(T\in\mathcal B(H)\),
(iii) there exist two unitary operators \(U,V\in\mathcal B(H)\) such that \(\phi(T)=UTV\) for all \(T\in\mathcal B(H)\).
This generalizes the result of M.Mbekhta [“Linear maps preserving the minimum and surjectivity moduli of operators” (preprint), Theorem 3.1] to the non-unital case.
MSC:
| 47B49 | Transformers, preservers (linear operators on spaces of linear operators) |
| 47A10 | Spectrum, resolvent |
Keywords:
minimum modulus; surjectivity modulus; left spectrum; right spectrum; unitary operators; linear preserversReferences:
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