Skip to main content Accessibility help
×
Home

Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

  • Abdellatif Bourhim (a1) and Constantin Costara (a2)

Abstract

In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector
      Available formats
      ×

Copyright

References

Hide All
[1] Aiena, P., Fredholm and local spectral theory, with applications to multipliers . Kluwer Academic Publishers, Dordrecht, 2004.
[2] Alaminos, J., Brešar, M., Šemrl, P., and Villena, A. R., A note on spectrum-preserving maps . J. Math. Anal. Appl. 387(2012), 595603. https://doi.org/10.1016/j.jmaa.2011.09.024.
[3] Alaminos, J., Extremera, J., and Villena, A. R., Approximately spectrum-preserving maps . J. Funct. Anal. 261(2011), 233266. https://doi.org/10.1016/j.jfa.2011.02.020.
[4] Alaminos, J., Brešar, M., Extremera, J., and Villena, A. R., Maps preserving zero products . Studia Math. 193(2009), 131159. https://doi.org/10.4064/sm193-2-3.
[5] Aupetit, B., Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras . J. London Math. Soc. 62(2000), 917924. https://doi.org/10.1112/S0024610700001514.
[6] Aupetit, B., Sur les transformations qui conservent le spectre. In: Banach algebras 97 (Blaubeuren), de Gruyter, Berlin, 1998, 55–78.
[7] Aupetit, B. and Mouton, H. T., Spectrum preserving linear mappings in Banach algebras . Studia Math. 109(1994), 91100. https://doi.org/10.4064/sm-109-1-91-100.
[8] Baribeau, L. and Ransford, T., Non-linear spectrum-preserving maps . Bull. London Math. Soc. 32(2000), 814. https://doi.org/10.1112/S0024609399006426.
[9] Bhatia, R., Šemrl, P., and Sourour, A., Maps on matrices that preserve the spectral radius distance . Studia Math. 134(1999), 99110.
[10] Botta, P., Pierce, S., and Watkins, W., Linear transformations that preserve the nilpotent matrices . Pacific J. Math. 104(1983), 3946. https://doi.org/10.2140/pjm.1983.104.39.
[11] Bourhim, A. and Mabrouk, M., Jordan product and local spectrum preservers . Studia Math. 234(2016), 97120.
[12] Bourhim, A. and Mabrouk, M., Maps preserving the local spectrum of Jordan product of matrices . Linear Algebra Appl. 484(2015), 379395. https://doi.org/10.1016/j.laa.2015.06.034.
[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. https://doi.org/10.1090/conm/638/12810.
[14] Bourhim, A. and Mashreghi, J., Maps preserving the local spectrum of product of operators . Glasgow Math. J. 57(2015), 709718. https://doi.org/10.1017/S0017089514000585.
[15] Bourhim, A. and Mashreghi, J., Maps preserving the local spectrum of triple product of operators . Linear Multilinear Algebra 63(2015), 765773. https://doi.org/10.1080/03081087.2014.898299.
[16] Bourhim, A. and Mashreghi, J., Local spectral radius preservers . Integral Equations Operator Theory 76(2013), 95104. https://doi.org/10.1007/s00020-013-2041-9.
[17] Bourhim, A., Burgos, M., and Shulman, V. S., Linear maps preserving the minimum and reduced minimum moduli . J. Funct. Anal. 258(2010), 5066. https://doi.org/10.1016/j.jfa.2009.10.003.
[18] Bourhim, A. and Miller, V., Linear maps on ${\mathcal{M}}_{n}(\mathbb{C})$ preserving the local spectral radius. Studia Math. 188 (2008), 67–75. https://doi.org/10.4064/sm188-1-4.
[19] Bourhim, A. and Ransford, T., Additive maps preserving local spectrum . Integral Equations Operator Theory 55(2006), 377385. https://doi.org/10.1007/s00020-005-1392-2.
[20] Bračič, J. and Müller, V., Local spectrum and local spectral radius of an operator at a fixed vector . Studia Math. 194(2009), 155162. https://doi.org/10.4064/sm194-2-3.
[21] Brešar, M. and Šemrl, P., Linear maps preserving the spectral radius . J. Funct. Anal. 142(1996), 360368. https://doi.org/10.1006/jfan.1996.0153.
[22] Costara, C., Automatic continuity for linear surjective maps compressing the local spectrum at fixed vectors, Proc. Amer. Math. Soc., 145, No. 5, (2017) 2081–2087. https://doi.org/10.1090/proc/13364.
[23] Costara, C., Surjective maps on matrices preserving the local spectral radius distance . Linear Multilinear Algebra 62(2014), 988994. https://doi.org/10.1080/03081087.2013.801967.
[24] Costara, C., Linear maps preserving operators of local spectral radius zero . Integral Equations Operator Theory 73(2012), 716. https://doi.org/10.1007/s00020-012-1953-0.
[25] Costara, C., Maps on matrices that preserve the spectrum . Linear Algebra Appl. 435(2011), 26742680. https://doi.org/10.1016/j.laa.2011.04.026.
[26] Costara, C., Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector . Arch. Math. 95(2010), 567573. https://doi.org/10.1007/s00013-010-0191-4.
[27] Costara, C. and Repovš, D., Nonlinear mappings preserving at least one eigenvalue . Studia Math. 200(2010), 7989. https://doi.org/10.4064/sm200-1-5.
[28] Dieudonné, J., Sur une généralisation du groupe orthogonal a quatre variables . Arch. Math. 1(1949), 282287. https://doi.org/10.1007/BF02038756.
[29] Flanders, H., On spaces of linear transformations with bounded rank . J. London Math. Soc. 37(1962), 1016. https://doi.org/10.1112/jlms/s1-37.1.10.
[30] Frobenius, G., Ueber die Darstellung der endlichen Gruppen durch lineare Substitutionen . Berl. Ber. (1897), 9941015.
[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), 10491069. https://doi.org/10.1016/j.laa.2009.10.018.
[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), 3147. https://doi.org/10.4064/sm184-1-2.
[33] Jafarian, A. A. and Sourour, A. R., Spectrum-preserving linear maps . J. Funct. Anal. 66(1986), 255261. https://doi.org/10.1016/0022-1236(86)90073-X.
[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.
[35] Marcus, M. and Moyls, B. N., Linear transformations on algebras of matrices . Canad. J. Math. 11(1959), 6166. https://doi.org/10.4153/CJM-1959-008-0.
[36] Miller, T. L., Miller, V. G., and Neumann, M. M., Local spectral properties of weighted shifts . J. Operator Theory 51(2004), 7188.
[37] Molnár, L. and Barczy, M., Linear maps on the space of all bounded observables preserving maximal deviation . J. Funct. Anal. 205(2003), 380400. https://doi.org/10.1016/S0022-1236(03)00213-1.
[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), 248262. https://doi.org/10.1006/jfan.2002.3970.
[39] Šemrl, P., Linear maps that preserve the nilpotent operators . Acta Sci. Math. 61(1995), 523534.
[40] Sourour, A. R., Invertibility preserving linear maps on  ${\mathcal{L}}(X)$ . Trans. Amer. Math. Soc. 348 (1996), 13–30. https://doi.org/10.1090/S0002-9947-96-01428-6.
[41] Torgašev, A., On operators with the same local spectra . Czechoslovak Math. J. 48(1998), 7783. https://doi.org/10.1023/A:1022467611697.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

  • Abdellatif Bourhim (a1) and Constantin Costara (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed