On the pseudospectrum preservers

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-2020-06-0089

Keywords:

Additive maps, Pseudospectrum preservers, Generalized products

Abstract

Let X and Y be two complex Banach spaces, and let B(X) denotes the algebra of all bounded linear operators on X. We characterize additive maps from B(X) onto B(Y ) compressing the pseudospectrum subsets ??(.), where ?? (.) stands for any one of the spectral functions ?? (.), ?l? (.) and ?r?  (.) for some ? > 0. We also characterize the additive (resp. non-linear) maps from B(X) onto B(Y) preserving the pseudospectrum ?? (.) of generalized products of operators for some ? > 0 (resp. for every ? > 0).

Author Biographies

Mustapha Ech-Chérif El Kettani, Sidi Mohammed Ben Abdellah University.

Dept. of Mathematics, LaSMA Laboratory.

aziz lahssaini, University Sidi Mohammed Ben Abdellah.

Dept. of Mathematics, LaSMA Laboratory.

References

Z. E. A. Abdelali, A. Achchi, and R. Marzouki, “Maps preserving the local spectral radius zero of generalized product of operators”, Linear and multilinear algebra, vol. 67, no. 10, pp. 2021–2029, Jul. 2018, doi: 10.1080/03081087.2018.1479371

B. Aupetit, “Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras”, Journal of the London mathematical society, vol. 62, no. 3, pp. 917–924, Dec. 2000, doi: 10.1112/S0024610700001514

H. Benbouziane, Y. Bouramdane, and M. Ech-Chérif El Kettani, “Maps preserving local spectral subspaces of generalised product of operators”, Rendiconti del Circolo Matematico di Palermo Series 2, vol. 69, no. 3, pp. 1033–1042, Dec. 2019, doi: 10.1007/s12215-019-00453-w

A. Bourhim, “Additive maps preserving the reduced minimum modulus of Banach space operators”, Journal of operator theory, vol. 67, no. 1, pp. 279-288, 2012. [On line]. Available: https://bit.ly/34CA0n1

M. Ech-Chérif El Kettani and H. Benbouziane, “Additive maps preserving operators of inner local spectral radius zero”, Rendiconti del Circolo Matematico di Palermo, vol. 63, no. 2, pp. 311–316, Aug. 2014, doi: 10.1007/s12215-014-0160-z

J. Hou, C.-K. Li, and N.-C. Wong, “Jordan isomorphisms and maps preserving spectra of certain operator products”, Studia mathematica, vol. 184, no. 1, pp. 31–47, 2008, doi: 10.4064/sm184-1-2

J. Hou, C.-K. Li, and N.-C. Wong, “Maps preserving the spectrum of generalized Jordan product of operators”, Linear algebra and its applications, vol. 432, no. 4, pp. 1049–1069, Feb. 2010, doi: 10.1016/j.laa.2009.10.018

J. Hou and L. Huang, “Additive maps between standard operator algebras compressing certain spectral functions”, Acta mathematica sinica, english series, vol. 24, no. 12, pp. 2041–2048, Dec. 2008., Dec. 2008, doi: 10.1007/s10114-008-6428-5

G. K. Kumar and S. H. Kulkarni, “Linear maps preserving pseudospectrum and condition spectrum”, Banach journal of mathematical analysis, vol. 6, no. 1, pp. 45-60, 2012, doi: 10.15352/bjma/1337014664

B. Kuzma, “Additive spectrum compressors”, Journal of mathematical analysis and applications, vol. 304, no. 1, pp. 13-21, Apr. 2005, doi: 10.1016/j.jmaa.2004.09.004

A. Krishnan and S. H. Kulkarni, “Pseudo spectrum of element in Banach algebra”, Operators and matrices, vol. 11, no. 1, pp. 263-287, Mar. 2017, doi: 10.7153/oam-11-18

S. Ragoubi, “On linear maps preserving certain pseudospectrum and condition spectrum subsets”. Advances in operator theory, vol. 3, no. 2, pp. 423-432, 2018, doi: 10.15352/AOT.1705-1159

H. Skhiri, “Reduced minimum modulus preserving in Banach space”, Integral equations and operator theory, vol. 62, no. 1, pp. 137-148, Sep. 2008, doi: 10.1007/s00020-008-1612-7.

L. N. Trefethen and M. Embree, Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton, NJ: Princeton University Press, 2005.

Published

2020-11-12

How to Cite

[1]
M. Ech-Chérif El Kettani and aziz lahssaini, “On the pseudospectrum preservers”, Proyecciones (Antofagasta, On line), vol. 39, no. 6, pp. 1457-1469, Nov. 2020.

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