Geodesic
Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector
Canadian journal of mathematics, Tome 71 (2019) no. 4, pp. 749-771

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.
DOI : 10.4153/CJM-2018-017-0
Mots-clés : linear preserver, local spectrum, local spectral radius, matrix 
Bourhim, Abdellatif; Costara, Constantin. Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector. Canadian journal of mathematics, Tome 71 (2019) no. 4, pp. 749-771. doi: 10.4153/CJM-2018-017-0
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     journal = {Canadian journal of mathematics},
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     year = {2019},
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-017-0/}
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[1] Aiena, P., Fredholm and local spectral theory, with applications to multipliers . Kluwer Academic Publishers, Dordrecht, 2004. Google Scholar

[2] Alaminos, J., Brešar, M., Šemrl, P., and Villena, A. R., A note on spectrum-preserving maps . J. Math. Anal. Appl. 387(2012), 595–603. . Google Scholar | DOI

[3] Alaminos, J., Extremera, J., and Villena, A. R., Approximately spectrum-preserving maps . J. Funct. Anal. 261(2011), 233–266. . Google Scholar | DOI

[4] Alaminos, J., Brešar, M., Extremera, J., and Villena, A. R., Maps preserving zero products . Studia Math. 193(2009), 131–159. . Google Scholar | DOI

[5] Aupetit, B., Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras . J. London Math. Soc. 62(2000), 917–924. . Google Scholar | DOI

[6] Aupetit, B., Sur les transformations qui conservent le spectre. In: Banach algebras 97 (Blaubeuren), de Gruyter, Berlin, 1998, 55–78. Google Scholar

[7] Aupetit, B. and Mouton, H. T., Spectrum preserving linear mappings in Banach algebras . Studia Math. 109(1994), 91–100. . Google Scholar | DOI

[8] Baribeau, L. and Ransford, T., Non-linear spectrum-preserving maps . Bull. London Math. Soc. 32(2000), 8–14. . Google Scholar | DOI

[9] Bhatia, R., Šemrl, P., and Sourour, A., Maps on matrices that preserve the spectral radius distance . Studia Math. 134(1999), 99–110. Google Scholar

[10] Botta, P., Pierce, S., and Watkins, W., Linear transformations that preserve the nilpotent matrices . Pacific J. Math. 104(1983), 39–46. . Google Scholar | DOI

[11] Bourhim, A. and Mabrouk, M., Jordan product and local spectrum preservers . Studia Math. 234(2016), 97–120. Google Scholar

[12] Bourhim, A. and Mabrouk, M., Maps preserving the local spectrum of Jordan product of matrices . Linear Algebra Appl. 484(2015), 379–395. . Google Scholar | DOI

[13] Bourhim, A. and Mashreghi, J., A survey on preservers of spectra and local spectra. In: Invariant subspaces of the shift operator, Contemp. Math., 638, American Mathematical Society, Providence, RI, 2015, pp. 45–98. . Google Scholar | DOI

[14] Bourhim, A. and Mashreghi, J., Maps preserving the local spectrum of product of operators . Glasgow Math. J. 57(2015), 709–718. . Google Scholar | DOI

[15] Bourhim, A. and Mashreghi, J., Maps preserving the local spectrum of triple product of operators . Linear Multilinear Algebra 63(2015), 765–773. . Google Scholar | DOI

[16] Bourhim, A. and Mashreghi, J., Local spectral radius preservers . Integral Equations Operator Theory 76(2013), 95–104. . Google Scholar | DOI

[17] Bourhim, A., Burgos, M., and Shulman, V. S., Linear maps preserving the minimum and reduced minimum moduli . J. Funct. Anal. 258(2010), 50–66. . Google Scholar | DOI

[18] Bourhim, A. and Miller, V., Linear maps on preserving the local spectral radius. Studia Math. (2008), 67–75. . Google Scholar | DOI

[19] Bourhim, A. and Ransford, T., Additive maps preserving local spectrum . Integral Equations Operator Theory 55(2006), 377–385. . Google Scholar | DOI

[20] Bračič, J. and Müller, V., Local spectrum and local spectral radius of an operator at a fixed vector . Studia Math. 194(2009), 155–162. . Google Scholar | DOI

[21] Brešar, M. and Šemrl, P., Linear maps preserving the spectral radius . J. Funct. Anal. 142(1996), 360–368. . Google Scholar | DOI

[22] Costara, C., Automatic continuity for linear surjective maps compressing the local spectrum at fixed vectors, Proc. Amer. Math. Soc., , No. 5, (2017) 2081–2087. . Google Scholar | DOI

[23] Costara, C., Surjective maps on matrices preserving the local spectral radius distance . Linear Multilinear Algebra 62(2014), 988–994. . Google Scholar | DOI

[24] Costara, C., Linear maps preserving operators of local spectral radius zero . Integral Equations Operator Theory 73(2012), 7–16. . Google Scholar | DOI

[25] Costara, C., Maps on matrices that preserve the spectrum . Linear Algebra Appl. 435(2011), 2674–2680. . Google Scholar | DOI

[26] Costara, C., Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector . Arch. Math. 95(2010), 567–573. . Google Scholar | DOI

[27] Costara, C. and Repovš, D., Nonlinear mappings preserving at least one eigenvalue . Studia Math. 200(2010), 79–89. . Google Scholar | DOI

[28] Dieudonné, J., Sur une généralisation du groupe orthogonal a quatre variables . Arch. Math. 1(1949), 282–287. . Google Scholar | DOI

[29] Flanders, H., On spaces of linear transformations with bounded rank . J. London Math. Soc. 37(1962), 10–16. . Google Scholar | DOI

[30] Frobenius, G., Ueber die Darstellung der endlichen Gruppen durch lineare Substitutionen . Berl. Ber. (1897), 994–1015. Google Scholar

[31] Hou, J. C., Li, C. K., and Wong, N. C., Maps preserving the spectrum of generalized Jordan product of operators . Linear Algebra Appl. 432(2010), 1049–1069. . Google Scholar | DOI

[32] Hou, J. C., Li, C. K., and Wong, N. C., Jordan isomorphisms and maps preserving spectra of certain operator products . Studia Math. 184(2008), 31–47. . Google Scholar | DOI

[33] Jafarian, A. A. and Sourour, A. R., Spectrum-preserving linear maps . J. Funct. Anal. 66(1986), 255–261. . Google Scholar | DOI

[34] Laursen, K. B. and Neumann, M. M., An introduction to local spectral theory. London Mathematical Society Monographs, New Series, 20, The Clarendon Press, Oxford University Press, New York, 2000. Google Scholar

[35] Marcus, M. and Moyls, B. N., Linear transformations on algebras of matrices . Canad. J. Math. 11(1959), 61–66. . Google Scholar | DOI

[36] Miller, T. L., Miller, V. G., and Neumann, M. M., Local spectral properties of weighted shifts . J. Operator Theory 51(2004), 71–88. Google Scholar

[37] Molnár, L. and Barczy, M., Linear maps on the space of all bounded observables preserving maximal deviation . J. Funct. Anal. 205(2003), 380–400. . Google Scholar | DOI

[38] Molnár, L., Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn’s version of Wigner’s theorem . J. Funct. Anal. 194(2002), 248–262. . Google Scholar | DOI

[39] Šemrl, P., Linear maps that preserve the nilpotent operators . Acta Sci. Math. 61(1995), 523–534. Google Scholar

[40] Sourour, A. R., Invertibility preserving linear maps on . Trans. Amer. Math. Soc. (1996), 13–30. . Google Scholar | DOI

[41] Torgašev, A., On operators with the same local spectra . Czechoslovak Math. J. 48(1998), 77–83. . Google Scholar | DOI

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