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Lipschitz spectrum preserving mappings on algebras of matrices. (English) Zbl 0819.15002

The author considers Lipschitz mappings \(T : M_ n (\mathbb{C}) \to M_ n (\mathbb{C})\) satisfying \(T(0) = 0\) and \(\sigma (T(A) - T(B)) \subset \sigma (A - B)\), for all \(A,B \in M_ n (\mathbb{C})\), where \(\sigma (C)\) is the spectrum of a matrix \(C\). It is shown that either \(T(A) = UAU^{-1}\) for all matrices \(A \in M_ n (\mathbb{C})\) or \(T(A) = UA^ t U^{-1}\) for all \(A\).

MSC:

15A04 Linear transformations, semilinear transformations
15A30 Algebraic systems of matrices
15A18 Eigenvalues, singular values, and eigenvectors
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References:

[1] Kowalski, S.; Slodkowski, Z., A characterization of multiplicative linear functionals in Banach algebras, Studia Math., 67, 215-223 (1980) · Zbl 0456.46041
[2] Marcus, M.; Moyls, B. N., Linear transformations on algebras of matries, Canad. J. Math., 11, 61-66 (1959) · Zbl 0086.01703
[3] Marcus, M.; Purves, R., Linear transformations on algebras of matrices, the invariance of the elementary symmetric functions, Canad. J. Math., 11, 383-396 (1959) · Zbl 0086.01704
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