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Preservers of local spectra of matrix sums. (English) Zbl 1282.47054

Let \(M_n(\mathbb{C})\) be the algebra of all \(n\times n\) complex matrices and \(e\) a nonzero vector of \(\mathbb{C}^n\). For a matrix \(T\in M_n(\mathbb{C})\), let \(\sigma_T(x)\) stand for the local spectrum of \(T\) at \(e\). Among other results established in [M. Bendaoud et al., Linear Multilinear Algebra 61, No. 7, 871–880 (2013; Zbl 1278.47042)], it is shown that a surjective map \(\varphi\) on \(M_n(\mathbb{C})\) satisfies \(\varphi(0)=0\) and \[ \sigma_{\varphi(T)-\varphi(S)}(e)\subset\sigma_{T-S}(e)\;(T,S\in M_n(\mathbb{C})) \] if and only if there exists an invertible matrix \(A\in M_n(\mathbb{C})\) such that \(Ae=e\) and \(\varphi(T)=ATA^{-1}\) for all \(T\in M_n(\mathbb{C})\). The proof uses a density argument and the automatic continuity of \(\varphi\) that follows an argument quoted from [C. Costara, Linear Algebra Appl. 435, No. 11, 2674–2680 (2011; Zbl 1231.15003)]. Furthermore, maps on \( M_n(\mathbb{C})\) verifying the reverse of the above inclusion are also characterized. In the paper under review, the author re-establishes similar results to the ones from the first cited paper, using sums of matrices instead of subtractions. The proofs are essentially the same.

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A53 (Semi-) Fredholm operators; index theories
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[1] Aiena, P., Fredholm and Local Spectral Theory, with Applications to Multipliers (2004), Kluwer Academic Publishers · Zbl 1077.47001
[2] Aupetit, B., A Primer on Spectral Theory (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0715.46023
[3] Bendaoud, M.; Sarih, M., Additive local spectrum compressors, Linear Algebra Appl., 435, 6, 1473-1478 (2011) · Zbl 1231.47031
[4] Bendaoud, M.; Sarih, M., Locally spectrally bounded linear maps, Math. Bohem., 136, 1, 81-89 (2011) · Zbl 1216.47066
[5] M. Bendaoud, M. Douimi, M. Sarih, Maps on matrices preserving local spectra, Linear and Multilinear Algebra, (2012), http://dx.doi.org/10.1080/03081087.2012.716429. · Zbl 1278.47042
[6] Bhatia, R.; Šemrl, P.; Sourour, A. R., Maps on matrices that preserve the spectral radius distance, Studia Math., 134, 99-110 (1999) · Zbl 0927.15006
[7] Bourhim, A.; Miller, V. G., Linear maps on \(M_n ( \mathbb{C} )\) preserving the local spectral radius, Studia Math., 188, 67-75 (2008) · Zbl 1145.47005
[8] Bračič, J.; Müller, V., Local spectrum and local spectral radius at a fixed vector, Studia Math., 194, 155-162 (2009) · Zbl 1182.47004
[9] Chan, J. T.; Li, C. K.; Sze, N. S., Mappings preserving spectra of products of matrices, Proc. Amer. Math. Soc., 135, 977-986 (2007) · Zbl 1113.15002
[10] Costara, C., Automatic continuity for surjective linear mappings decreasing the local spectral radius at some fixed vector, Arch. Math., 95, 567-573 (2010) · Zbl 1210.47016
[11] González, M.; Mbekhta, M., Linear maps on \(M_n ( \mathbb{C} )\) preserving the local spectrum, Linear Algebra Appl., 427, 2-3, 176-182 (2007) · Zbl 1127.15005
[12] Jafarian, A.; Sourour, A. R., Spectrum preserving linear maps, J. Funct. Anal., 66, 255-261 (1986) · Zbl 0589.47003
[13] Laursen, K. B.; Neumann, M. M., An Introduction to Local Spectral Theory (2000), Oxford University Press: Oxford University Press New York · Zbl 0957.47004
[14] Müller, V., Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Operator Theory: Advances and Applications, vol. 139 (2007), Birkhäuser · Zbl 1208.47001
[15] Sourour, A. R., Inversibility preserving linear maps on \(\mathcal{L} ( X )\), Trans. Amer. Math. Soc., 348, 13-30 (1996) · Zbl 0843.47023
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